












































































































































































































































































































































Class 

Book 






Copyright N°_ 

COPYRIGHT DEPOSIT. 





































































■ 






















































































































































































WORKS OF H. M. WILSON 

PUBLISHED BY 

JOHN WILEY & SONS. 


Topographic Surveying. 

Including geographic, exploratory, and military 
mapping, with hints on camping, emergency surgery, 
and photography. Third Edition, Revised. Illus¬ 
trated by 18 engraved colored plates and 205 half-tone 
plates and cuts, including two double-page plates. 
8vo, xxx + 910 pages. Cloth, $3.50. 

Irrigation Engineering. 

Part I. Hydrography. Part II. Canals and 
Canal Works. Part III. Storage Reservoirs. 
Fifth Edition, Revised. Small 8vo, xxx-)-614 pages; 
41 full-page plates and 139 figures, including many half¬ 
tones. $4.00. 










[ Frontispiece.] 



Plate I.—Surveying under Difficulties 






TOPOGRAPHIC SURVEYING. 


INCLUDING 

GEOGRAPHIC, EXPLORATORY, AND 
MILITARY MAPPING, 

WITH HINTS ON 

CAMPING, EMERGENCY SURGERY, AND 
PHOTOGRAPHY. 


HERBERT M. WILSON, 

Geographer, United States Geological Survey ) Member American Society of 
Civil Engineers ; Author of “ Irrigation Engineering,” etc. 


THIRD EDITION, DEVISED. 
FIRST THOUSAND. 


NEW YORK: 

JOHN WILEY & SONS. 

London : CHAPMAN & HALL, Limited. 

1908. 

v;.-, 7 77-r/J ;nrn:i‘! ,d-'f VWJ-VI 


« 

'J 



f’ .♦ *'1 f • f, 

• A ^ 


LIBRARY of CONGRESS 1 

Two Copies KeceivviG 

MAR 10 1908 

OopynKm C-iUrj* 

^6 /<?<># 

CLASS ^ XXc. No, 

^73 

OPY B. 



Copyright, 1900, 1905, 1908, 

BY 

HERBERT M. WILSON. 




ROBERT DRUMMOND, PRINTER, NEW YORK, 






PREFACE. 




This book has been prepared with a view of bringing 
together in one volume the data essential to a comprehensive 
knowledge of topographic surveying. It has been my aim 
to cover the varied phases of all classes of surveys which are 
made with a view to representing on maps information rela¬ 
tive to the features of the earth’s surface. The methods 
elaborated are chiefly those which have been developed in 
recent years by the great government surveying organiza¬ 
tions and by such few private corporations as have kept in 
touch with the most modern practice; but I have endeavored 
to go beyond these, and, guided by personal experience, to 
adapt them to the most detailed topographic as well as to 
the crudest exploratory surveys. The hope is entertained, 
therefore, that the engineer who may be called upon to con¬ 
duct an exploratory survey in an unknown region, or to make 
a detailed topographic map as a preliminary to construction, 
will find herein descriptions and examples of the methods he 
should employ, the essential tables for the computation of 
his results, and hints which will guide in the equipment of his 
party. 

I have sought to avoid any detailed description of those 
instruments or methods which are elaborated in works on 
general surveying. The volume is devoted practically to 
higher surveying, and presupposes a knowledge of all the 
more elementary branches. At the same time, many of the 



IV 


PREFA CE. 


subjects treated are essentially elementary, and these are 
briefly described, in order that all the facts which the topog¬ 
rapher must know and all the formulas and tables which he 
must have at hand in the field may be brought together. An 
effort has been made to present the subject in the most prac¬ 
tical form. Accordingly, care has been taken to avoid an 
elaboration of the mathematical processes by which the vari¬ 
ous formulas have been derived, as they are to be found in 
detail in several well-known treatises to which textual refer¬ 
ence is made. To give more immediate aid to the working 
surveyor, examples of the various computations are pre¬ 
sented, as are illustrations of the instruments, methods, and 
resulting maps from surveys actually executed. 

The mode of presentation is not that usually followed in 
such works. Instead of describing the instruments or their 
uses independently, each is described in that portion of the 
text in which its employment in field surveying is most prom¬ 
inently mentioned. The tables are not brought together at 
the end of the volume, but each is placed in that portion of the 
text which relates to its use. The object is to produce a handy 
reference-book for use in the field, as well as a text-book for 
guidance in college instruction. It is believed that, by this 
arrangement, if a topographer in the midst of his field-work 
desires information on a special point, it can be found, with 
accompanying examples and tables, gathered together in one 
chapter or clearly indicated by cross-references. Again, the 
method of treatment usually followed in works of this class 
consists in, first, a description of the astronomic methods on 
which general map surveys must be based, and then a descrip¬ 
tion of primary triangulation as a basis for the detailed 
topographic surveys which are finally described. I have re- 

, , * f • - * - , 

versed this order and have adopted the more natural method 
of commencing with the simplest operations and advancing 
gradually towards the most complex and refined. Each sub¬ 


ject is treated ift the same manner. It is believed that the 


m 


PREFACE. 


V 


work has thus been made especially useful to the inexpert 
topographer and the student. 

The volume consists, in fact, of three separate books or 
treatises: (i) Topographic Surveying, (2) Geodetic Survey¬ 
ing, and (3) Practical Astronomy. The first has been sub¬ 
divided into three parts: Plane Surveying, Hypsometric Sur¬ 
veying, and Map Construction; and these are preceded by a 
preliminary characterization of the relations existing between 
topographic, geographic, and exploratory surveys. This latter 
distinction is essentially arbitrary, as all are of a kind, and 
differ only in degree of detail and the consequent speed and 
generalization in procuring the field results. The general 
subject of Geodetic Surveying has been subdivided into Ter¬ 
restrial Geodesy and Astronomic Geodesy, and the treatment 
of these differs but slightly in method of arrangement from 
that usually pursued. Part VII is devoted to such practical 
hints as it is believed will essentially aid those who have the 
organization and command of camping parties. 

I am especially indebted to the courtesy of Professors 
Ira O. Baker, J. B. Johnson, and John F. Hayford for the 
use of numerous electrotypes and plates from their well- 
known works on surveying and geodesy; and to the Secre¬ 
tary of the American Society of Civil Engineers for electro¬ 
types of illustrations in articles by me. I am also indebted to 
Messrs. W. & L. E. Gurley, Young & Sons, and G. N. Saeg- 
muller for electrotypes of instruments illustrated in their cata¬ 
logues. I have used freely the excellent Manual of Topo¬ 
graphic Methods of the U. S. Geological Survey, written by 
Mr. Henry Gannett; in a few instances I have copied verba¬ 
tim examples contained therein, and I desne to express appie 
ciation of his courtesy, and of that of the Diiectoi of the U. S. 
Geological Survey in extending this privilege. To the latter 
I am also indebted for an opportunity to procure the colored 
illustrations published herewith, which were printed from the 
admirable copper-plates of the IP'S. Geological Survey. Spe- 


VI 


PREFA CE. 


cifications and several illustrations of tents and other camp 
equipage were obtained through the courtesy of the Quarter¬ 
master-General of the U. S. Army. For much in the chapter 
on Photography I am indebted to Lieut. Samuel Reber's 
Manual of Photography and to E. Deville's Photographic 
Surveying. 

Finally, I desire to express appreciation of the assistance 
I have received in editing manuscript and proof from many 
coworkers on the U. S. Geological Survey, more particularly 
from Messrs. W. J. Peters, S. S. Gannett, and E. M. Douglas 
on the subjects of geodesy and astronomy; E. C. Barnard 
and A. H. Thompson on topographic surveying; C. Willard 
Hayes and G. K. Gilbert on topographic forms and definitions; 
N. H. Darton on photography; and to Mr. W. Carvel Hall 
for assistance in reading proof. Two lists of works of refer¬ 
ence are published, on pages 490 and 809, in which are cited 
the titles of all those works to which the reader is referred for 
further details. From nearly all of these some example or 
illustration has been obtained. 

H. M. W. 

Washington, D. C., Feb. 22, 1900. 


PREFACE TO SECOND EDITION. 


In preparing this second edition no radical changes have 
been made in any chapter. Numerous minor changes and 
corrections have been made, however, chiefly in the nature of 
citations of new practices or correction of old. This is par¬ 
ticularly true of the subject of Projections, which is better illus¬ 
trated, that of Precise Leveling, and the bringing up to date of 
tables of Polaris Observations. 


Washington, D. C., Feb. 20, 1905. 


H. M. W. 




CONTENTS. 


PART I. 

TOPOGRAPHIC , GEOGRAPHIC , AND EXPLORATORY SURVEYING. 

CHAPTER I. 

KINDS OF MAP SURVEYS. 

ART. PAGE 

1. Classes of Surveys... i 

2. Information Surveys. 3 

3. Topographic Surveys. 4 

4. Features Shown on Topographic Maps. 5 

5. Public Uses of Topographic Maps. 6 

6. Degree of Accuracy Desirable in Topographic Surveys. 8 

7. Instructions Relative to Topographic Field-work. 12 

8. Elements of a Topographic Survey. 14 

CHAPTER II. 

SURVEYING FOR SMALL-SCALE OR GENERAL MAPS. 

9. Methods of Topographic Surveying. 18 

10. Geological Survey Method of Topographic Surveying. 20 

11. Organization of Field Survey. 22 

12. Surveying Open Country. 23 

13. Sketching Open Country. 28 

14. Surveying Woodland or Plains. 33 

15. Sketching Woodland or Plains. 35 

16. Control from Public Land Lines. 36 

17. Sketching Over Public Land Lines.37 

18. Cost of Topographic Surveys. 40 

19. The Art of Topographic Sketching. 40 

20. Optical Illusions in Sketching Topography. 44 

• • 

Vll 























Vlll 


CONTENTS. 


CHAPTER III. 

SURVEYING FOR DETAILED OR SPECIAL MAPS. 

ART. PAGE 

21. Topography for Railway Location. 47 

22. Detailed Topographic Surveys for Railway Location. 49 

23. Topographic Survey for Canal Location. 52 

24. Surveys for Reservoirs. 57 

25. Survey of Dam Site. r . 58 

26. City Surveys. 62 

27. Cadastral and Topographic City Survey. 64 

28. Cost of Large-scale Topographic Surveys. 67 

CHAPTER IV. 

£)VVf in5tVJc \VvU\K/.vJA-v7.A vivVR. 

GEOGRAPHIC AND EXPLORATORY SURVEYS. 

29. Geographic Surveys. 68 

30. Instrumental Methods Employed in Geographic Surveys. 69 

31. Geographic Maps. 70 

32. Features. Shown on Geographic Maps. 72 

33. Geographic Reports.,,.,,.. 73 

34. Scale and Cost of Governmental Geographic Surveys. 74 

35. Exploratory Surveys. 76 

36. Exploratory and Geographic Surveys Compared. 77 

37. Methods and Examples of Exploratory Surveys. 82 


CHAPTER V. 

MILITARY AND CADASTRAL SURVEYS. 

38. Military Surveys. 92 

39. Military Reconnaissance w r ith Guide Map. 95 

40. Military Reconnaissance without Guide Map. 95 

41. Detailed Military Map. 100 

42. Military Siege Maps. 101 

43. Military Sketches and Memoirs. 102 

44. Cadastral Surveys.. 103 

I? .‘ ^ 


TOPOGRAPHIC FORMS. 

45. Relations of Geology to Topography. 108 

46. Origin and Development of Topographic Forms. 109 

47. Physiographic- Processes. no 

48. Classification of Physiographic Processes. 112 





































COXTEXTS. 


IX 


30 -* Sjy 

page 

49. Erosion, Transportation, and Corrasion. 113 

50. Topographic Forms. 120 

51. Classification of Topographic Forms. 122 

Glossary of Topographic Forms . 133 


PART II. 

PLANE AND TACHYMETRIC SURVEYING. 

CHAPTER VII. 

PLANE-TABLES AND ALIDADES. 

52. Plane and Topographic Surveying. 146 

53. Plane-table Surveying. 147 

54. Reconnaissance and Execution of Plane-table Triangulation.... 149 

55. Tertian* Triangulation from Topographic Sketch Points. 151 

56. Varieties of Plane-tables. 152 

57. Plane-table Tripods and Boards. 153 

58. Plane-table Movements. 153 

59. Telescopic Alidades. 157 

60. Adjustments of Telescopic Alidade. 159 

61. Gannett Plane-table. 160 

62. Sight-alidades .. 161 

63. Folding Exploratory Plane-table. 163 

64. Cavalry Sketch-board. 164 

64a. Batson Sketching-case.166a 

CHAPTER VIII. 

SCALES. PLANE-TABLE PAPER. AND PENCILS. 

65. Scales. 166c 

65a. Special Scales..... 167 

66. Slide-rule ..*. 168 

67. Using the Slide-rule. 169 

68. Plane-tabie Paper. 174 

69. Preparation of Field Sheets. 175 

70. Manipulation of Pencil and Straightedge. 178 

71. Needle Points, Pencil Holders and Sharpeners. 179 

CHAPTER IX. 

PLANE-TABLE TRIANGULATION. 

72. Setting up the Plane-table. 180 

73. Location by Intersection. 182 
































8o. 

81 

82, 

83 

84 

85. 

86 , 

87 

88 , 

89 

90 

91. 

92, 

93 

94 

95 . 

96. 

97 . 

98, 

99, 


CONTENTS. 


PAGE 


Location by Resection. 185 

Three-point Problem Graphically Solved. 186 

Tracing-paper Solution of the Three-point Problem. 187 

Bessel’s Solution of the Three-point Problem. 188 

Coast Survey Solution of the Three-point Problem. 190 

Ranging-in, Lining-in, and Two-point Problem. 192 


CHAPTER X. 


TRAVERSE INSTRUMENTS AND METHODS. 


Traverse Surveys. 

Traversing by Plane-table and Magnetic Needle. 

Control by Large-scale Magnetic Traverse with Plane-table.... 

Traversing by Plane-table with Deflection Angles. 

Intersection from Traverse. 

Engineers’ Transit. 

Adjustments of the Transit. 

Traversing with Transit. 

Platting Transit Notes with Protractor and Scale. 

Protractors . 

Platting Transit Notes by Latitudes and Departures. 

Prismatic Compass. 

Magnetic Declination. 

Secular Variation and Annual Change. 

Local Attraction. 


195 

197 

199 

200 

202 

203 

204 
207 
210 
210 
212 
214 

221 

222 

223 


CHAPTER XI. 

LINEAR MEASUREMENT OF DISTANCES. 


Methods of Measuring Distances; Pacing. 224 

Distances by Time. 226 

Measuring Distance with Linen Tape. 228 

Odometer . 229 

Chaining . 234 


CHAPTER XII. 

STADIA TACHYMETRY. 

Tachymetry . 236 

Topography with Stadia. 237 

Tachymetry with Stadia. 238 

Accuracy and Speed of Stadia Tachymetry. 240 

Stadia Formula with Perpendicular Sight..... 241 

Stadia Formula with Inclined Sight. 246 

































CONTENTS. xi 

ART. PAGE 

106. Determining Horizontal Distances from Inclined Stadia Meas¬ 

ures . 249 

107. Horizontal Distances and Elevations from Stadia Readings_249 

108. Determining Elevations by Stadia. 258 

109. Diagram for Reducing Stadia Measures. 259 

no. Diagram for Reducing Inclined Stadia Distances to Horizontal.. 264 
in. Effects of Refraction on Stadia Measurements . 266 

112. Stadia-rods . 269 

CHAPTER XIII. 

ANGULAR TACHYMETRY. 

113. Angular i'achymetry with Transit or Theodolite. 272 

114. Measuring Distances with Gradienter. 274 

115. Wagner-Fennel Tachymeter. 280 

116. Range-finding . 282 

116a. Estimating Distances. 283 

117. Surveying with Range-finder.283 

118. Traversing with Range-finder. 284 

119. Weldon Range-finder. 285 

120. Accuracy and Difficulties of Range-finding. 289 

121. Range-finding with Plane-table. 290 

121a. Automatic Surveying Instruments.291 

CHAPTER XIV. 

PHOTOGRAPHIC SURVEYING. 

122. Photo-surveying . 29:.’ 

123. Photo-surveying and Plane-table Surveying Compared. 292 

124. Principles of Photo-topography. 296 

125. Camera and Plates. 298 

126. Field-work of a Photo-topographic Survey. 299 

127. Projecting the Photo-topographic Map. 300 


PART III. 

HYPSOMETRY, OR DETERMINATION OF HEIGHTS. 

CHAPTER XV. 

SPIRIT-LEVELING. 

128. Hypsometry . 305 

129. Spirit-leveling . 3°6 

130. Engineering Spirit-levels. 3°8 




























CONTENTS. 


Xll 

I>C 

ART. PAGE 

131. Adjustments of the Level. 308 

132. Target Leveling-rods........ 311 

133. Speaking-rods . 313 

134. Turning-points . 315 

135. Bench-marks . f 316 

136. Method of Running Single Lines of Levels. 317 

137. Instructions for Leveling. 320 

138. Note-books . 322 

139. Platting'Profiles. 324 

CHAPTER XVI. 

LEVELING OF PRECISION. 

140. Precise Leveling. 325. 

141. Binocular Precise Level. 326. 

142. Precise Spirit-level. 327 

143. Sequence in Simultaneous Double-rodded Leveling. 329 

144. Methods of Running. 332 

145. Precise Rods..... ..... 332 

146. Manipulation of Instrument. 336 

147. Length of Sight. 337 

148. Sources of Error. 339 

149. Divergence of Duplicate Level Lines. 343 

150. Limit of Precision.,. 344 

151. Adjustment of Group of Level Circuits. 345 

152. Refraction and Curvature. 347 

1,53. Speed in Leveling. 349, 

154. Cost of Leveling. 349 

155. Long-distance Precise Leveling. 352 

156. Hand-levels . 355 

157. Using the Locke Hand-level. 356 

158. Abney Clinometer Level. 357 


CHAPTER XVII. 

TRIGONOMETRIC LEVELING. 

159. Trigonometric Leveling. 359' 

160. Vertical Angulation i.. *.*... 361 

161. Vertical Angulation, Computation. 363 

162. Vertical Angulation in Sketching. 363 

163. Vertical Angulation from Traverse. :.... 36^ 

164. Trigonometric Leveling. Computation. 363 

165. Rrrp.r.s. in. Vertical. .Triangulati.on „ . 370 

166. Refraction, and Curvature, .. 371 

167; Leveling with .Gradients ..>... 372 










































CON 7'ENTS. 

^ Uu i - • 


Xlll 


CHAPTER XVIII. 

BAROMETRIC LEVELING. 

art. . 

168. Barometric Leveling..... 

169. Methods and Accuracy of Barometric Leveling 

170. Mercurial Barometer. . . . 

171. Barometric Notes and Computation. 

172. Example of Barometric Computation. 

173. Guyot’s Barometric Tables. 

174. Aneroid Barometer. 

175. Errors of Aneroid. 

176. Using the Aneroid. 

177. Thermometric Leveling..,... 




PAGE 

374 

375 

376 
37 & 
381 

383 

395 

395 

396 
402 


.V - 


PART IV. 

OFFICE WORK OF TOPOGRAPHIC MAPPING. 


CHAPTER XIX. 

MAP CONSTRUCTION. 

178. Cartography . 404 

179. Map Projection. 405 

180. Kinds of Projections. 405 

181. Perspective Projections. 405 

182. Cylinder Projections. 410 

183. Conical Projections. 412 

184. Constructing a Polyconic Projection. 416 

185. Projection of Maps upon a Polyconic Development. 418 

186. Use of Projection Tables. 435 

187. Areas of Quadrilaterals of Earth’s Surface. 436 

188. Platting Triangulation Stations on Projection. 437 

189. Scale Equivalents. 446 


CHAPTER XX. 

TOPOGRAPHIC DRAWING AND RELIEF MODELING. 


190. Methods of Map Construction. 449 

^ ..... ■ o 

191. Topographic Drawing. 449 

192. Contour Lines. 455' 

193. Contour Construction. 460 

; * • • • • •• • • •• • • • • •• • • •• •••*••• r -* *• 1 * •* J ■+ * l it. 1 - a. » lAlt . i -v 









































XIV 


CONTENTS . 


ART. PAGE 

194. Hachures . 461 

195. Conventional Signs. 463 

196. Lettering . 477 

197. Drafting Instruments. 477 

198. Model and Relief Maps. 478 

199. Modeling the Map. 480 

200. Duplicating the Model, Casting. 485 

Works of Reference on Topography. 490 


PART V. 

1 

TERRESTRIAL GEODESY. 

CHAPTER XXI. 

FIELD-WORK OF BASE MEASUREMENT. 

201. Geodesy . 495 

202. Base Measurement. 497 

203. Accuracy of Base Measurement. 498 

204. Base Measurement with Steel Tapes. 500 

205. Steel Tapes. 501 

206. Tape-stretchers . 501 

207. Laying out the Base. 505 

208. Measuring the Base. 507 

209. Compensated Base Bars. 507 

210. Contact-slide Base Apparatus. 508 

211. Iced-bar Apparatus. 511 

212. Repsold Base Apparatus. 514 

213. Base Lines: Cost, Speed, and Accuracy. 516 

CHAPTER XXII. 

COMPUTATION OF BASE MEASUREMENT. 

214. Reduction of Base Measurement. 517 

215. Reduction to Standard. 517 

216. Correction for Temperature. 518 

217. Record of Base Measurement. 518 

218. Correction for Inclination of Base. 519 

219. Correction for Sag. 521 

220. Reduction of Base to Sea-level. 522 

221. Summary of Measures of Sections. 523 
































CONTENTS . XV 

ART * PAGE 

222. Corrected Length of Base. 523 

-223. Transfer of Ends of Base to Triangulation Signals. 524 

224. Other Corrections to Base Measurements. 526 

225. To Reduce Broken Base to Straight Line. 526 

CHAPTER XXIII. 

FIELD-WORK OF PRIMARY TRAVERSE. 

226. Traverse for Primary Control. 527 

227. Errors in Primary Traverse. 528 

228. Instruments Used in Primary Traverse. 524 

229. Method of Running Primary Traverse. 531 

230. Record and Reduction of Primary Traverse. 532 

231. Instructions for Primary Traverse. 533 

232. Cost, Speed, and Accuracy of Primary Traverse. 536 

CHAPTER XXIV. 

COMPUTATION OF PRIMARY TRAVERSE. 

233. Computation of Primary Traverse. 538 

234. Correction for Observed Check Azimuths. 539 

235. Computation of Latitudes and Longitudes. 540 

236. Corrected Latitudes and Longitudes. 542 

CHAPTER XXV. 

FIELD-WORK OF PRIMARY TRIANGULATION. 

237. Primary Triangulation. 545 

238. Reconnaissance for Primary Triangulation. 546 

239. Intervisibility of Triangulation Stations. 549 

240. Accuracy of Triangulation. 553 

241. Instruments . 553 

242. Micrometer Microscope. 556 

243. Triangulation Signals. 559 

244. Tripod and Quadripod Signals. 561 

245. Observing Scaffolds. 565 

246. Heliotrope . 566 

247. Night Signals. 574 

248. Station- and Witness-marks. 575 

CHAPTER XXVI. 

MEASUREMENT OF ANGLES. 

249. Precautions in Measuring Horizontal Angles. 577 

250. Observer’s Errors and their Correction. 578 































XVI 


CONTENTS. 


ART. PAGE 

251. Instrumental Errors and their Correction. 580 

252. Methods of Measuring Horizontal Angles. 584 

253. Record of Triangulation Observations. 588 

254. Instructions for Field-work of Primary Triangulation. 590 

255. Primary Triangulation—Cost, Speed, and Accuracy. 592 

CHAPTER XXVII. 

SOLUTION OF TRIANGLES. 

256. Trigonometric Functions. 594 

257. Fundamental Formulas for Trigonometric Functions. 594 

258. Formulas for Solution of Right-angled Triangles. 594 

259. Solution of Plane Triangles. 596 

260. Given Two Sides and Included Angle, to Solve the Triangle.... 598 

261. Given Certain Functions of a Triangle, to Find Remainder. 59S 

262. Given Three Sides of a Triangle, to Find the Angles. 599 

263. Three-point Problem. 600 

CHAPTER XXVIII. 

ADJUSTMENT OF PRIMARY TRIANGULATION. 

264. Method of Least Squares. 602 

265. Rejection of Doubtful Observations. 604 

266. Probable Error of Arithmetic Mean. 607 

267. Reduction to Center. 608 

268. Station Adjustment. 611 

269. Routine of Station Adjustment. 612 

270. Equations of Condition. 612 

271. Formation of Table of Correlates. 614 

272. Formation of Normal Equations and Substitution in Table of 

Correlates . 615 

273. Figure Adjustment. 616 

274. Routine of Figure Adjustment. 617 

275. Notation Used in Figure Adjustment. 618 

276. Angle Equations. 619 

277. Spherical Excess. 619 

278. Side Equations. 623 

279. Solution of Angle and Side Equations. 625 

280. Correlates and Normal Equations. 627 

281. Algebraic Solution of Normal Equations. 628 

282. Substitution in Normal Equations. 632 

283. Substitution in Table of Correlates. 632 

284. Weighted Observations. 633 



































CONTENTS. 


XVII 


CHAPTER XXIX. 

COMPUTATION OF DISTANCES AND COORDINATES. 

ART. PAGE 

285. Geodetic Coordinates. 636 

286. Computation of Distances. 637 

287. Formulas for Computing Geodetic Coordinates. 638 

288. Computation of Geodetic Coordinates: Example. 642 

289. Knowing Latitudes and Longitudes of Two Points, to Compute 

Azimuths and Distances. 646 

CHAPTER XXX. 

GEODETIC CONSTANTS AND REDUCTION TABLES. 

290. Constants Depending on Spheroidal Figure of Earth. 672 

291. Numerical Constants. 672 

292. Length of the Meter in Inches. 674 

293. Interconversion of English and Metric Measures. 675 

294. Logarithms and Factors for Conversion of English and Metric 

Measures . 676 


PART VI. 

GEODETIC ASTRONOMY. 

CHAPTER XXXI. 

ASTRONOMIC METHODS. 

295. Method of Treatment. 678 

296. Geodetic Astronomy. 679 

297. Definitions of Astronomic Terms. 679 

298. Astronomic Notation. 683 

299. Fundamental Astronomic Formulas. 684 

300. Finding the Stars. 686 

301. Parallax . 688 

302. Refraction . 689 

CHAPTER XXXII. 

TIME. 

303. Interconversion of Time. 69^ 

304. Interconversion of Time and Arc. 698 























XVtll 


CONTENTS. 


ART. PAGK 

305. Determination of Time. 7 00 

306. Time by a Single Observed Altitude of a Star. 702 

307. Approximate Time from Sun. 703 

308. Time by Meridian Transits. 703 

CHAPTER XXXIII. 

AZIMUTH. 

309. Determination of Azimuth. 707 

310. Observing for Azimuth. 707 

311. Approximate Solar Azimuth. 708 

312. Azimuths of Secondary Accuracy. 712 

313. Primary Azimuths. 719 

314. Reduction of Azimuth Observations. 720 

315. Azimuth at Elongation. 721 

CHAPTER XXXIV. 

LATITUDE. 

316. Methods of Determining Latitude. 723 

317. Approximate Solar Latitude. 724 

318. Latitude from an Observed Altitude. 725 

319. Astronomic Transit and Zenith Telescope. 726 

320. Latitude by Differences of Zenith Distances of Two Stars. 728 

321. Errors and Precision of Latitude Determinations. 729 

322. Field-work of Observing Latitude. 730 

323. Determination of Level and Micrometer Constants. 732 

324. Corrections to Observations for Latitude by Talcott’s Method.. 738 

325. Reduction of Latitude Observations. 743 

CHAPTER XXXV. 

LONGITUDE. 

326. Determination of Longitude. 744 

327. Astronomic Positions: Cost, Speed, and Accuracy. 744 

328. Longitude by Chronometers. 745 

329. Longitude by Lunar Distances. 746 

330. Longitude by Chronograph.:. 748 

331. Observing for Time. 751 

332. Reduction of Time Observations. 752 

333. Record of Time Observations. 754 

334. Longitude Computation. 757 

335 - Comparison of Time. 774 
































CONTENTS. 


XIX 


CHAPTER XXXVI. 

SEXTANT AND SOLAR ATTACHMENT. 

ART. PAGE 

336. Sextant . 777 

337. Adjustment of Sextant. 778 

338. Using the Sextant . 780 

339. Solar Attachment. 781 

340. Burt Solar Attachment. 781 

341. Adjustment of Burt Solar Attachment. 782 

342. Smith Meridian Attachment. 785 

343. Adjustment of Smith Meridian Attachment. 780 

344. Determination of Azimuth and Latitude with Solar Attachment. 789 

345. Solar Attachment to Telescopic Alidade. 791 

345a. Traversing with Solar Alidade. 792 

CHAPTER XXXVII. 

PHOTOGRAPHIC LONGITUDES. 

346. Field-work of Observing Photographic Longitude. 793 

347. The Camera and its Adjustments. 794 

348. Measurement of the Plate. 797 

349. Computation of the Plate. 801 

350. Sources of Error. 802 

351. Precision of Resulting Longitude. 806 

Reference Works on Geodesy . 809. 


PART VII. 

CAMPING, EMERGENCY SURGERY, PHOTOGRAPHY . 
CHAPTER XXXVIII. 

CAMP EQUIPMENT AND PROPERTY. 

3£2. Attributes of a Skillful Photographer. 81 r 

353. Subsistence and Transportation of Party in Field. 813 

354. Selecting and Preparing the Camp Ground. 814 

355. Tents . 817/ 

356. Specifications for Army Wall Tents. 820- 

357. Specifications for Army Wall-tent Flies. 821 

358. Specifications for Army Wall-tent Poles. 822 

359. Specifications for Army Shelter Tents (Halves). 822 

360. Specifications for Army Shelter Tents (Poles). 824 

361. Erecting the Tent. 825 






























XX 


CONTENTS. 


ART. PAGB 

362. Tent Ditching and Flooring. 825 

363. Camp Stoves, Cots, and Tables. 827 

364. Specifications for Army Sibley Tent Stoves. 829 

365. How to Build Camp-fires. 830 

366. Cooking-fire for a Small Camp. 830 

367. Camp Equipment. 831 

368. Provisions . 833 

CHAPTER XXXIX. 

TRANSPORTATION EQUIPMENT. 

369. Camp Transportation: Wagons. 836 

370. Pack Animals and Saddles. 837 

371. Moore Pack-saddles. 840 

372. Throwing the Diamond Hitch. 841 

373. Packmen . 847 

374. Transportation Repairs.'. 848 

375. Veterinary Surgery. 849 

CHAPTER XL. 

CARE OF HEALTH. 

376. Blankets and Clothing. 850 

377. Care of Health. 852 

378. Drinking-water . 855 

379. Medical Hints. 856 

380. Diarrhea and Dysentery. 857 

381. Drowning and Suffocation. 858 

382. Serpent- and Insect-bites. 860 

383. Surgical Advice. 860 

384. Medicine-chest . 861 

CHAPTER XLI. 

PHOTOGRAPHY. 

385. Uses of Photography in Surveying. 864 

386. Cameras . 865 

387. Lenses and their Accessories. 867 

388. Dry Plates and Films. 869 

389. Exposures . 872 

390. Developing . 875 

391. Fixing . 878 

392. Printing and Toning. 880 

303. Blue Prints and Black Prints. 883 


































TABLES. 


TABLE PAGE 

I. Scale and Cost of Detailed Topographic Maps. 67 

II. Scale, Cost, and Relief of Government Geographic Maps 75 

III. Scale and Cost of Cadastral Surveys. 107 

IV. Error in Horizontal Angle due to Inclination of Plane- 

table Board. 181 

V. Logarithms of Numbers to Four Places. 215 

VI. Logarithms of Trigonometric Functions. 217 

VII. For Converting Wheel-revolutions into Decimals of 

a Mile. 233 

VIII. Reduction of Inclined Stadia Measures to Horizontal 

Distances . 249 

IX. Horizontal Distances and Elevations from Stadia Read¬ 
ings . 250 

X. Differences of Elevation from Stadia Measures. 260 

XI. Natural Sines and Cosines.•.275 

XII. Natural Tangents and Cotangents. 277 

XIII. Cost of Leveling per Mile in Various States. 350 

XIV. Cost and Speed of Government Precise Leveling. 351 

XV. Differences of Altitude from Angles of Elevation or De¬ 
pression . 364 

XVI. Logarithms of Radius of Curvature R in Meters. 369 

XVII. Reduction of Barometric Readings to Feet. 384 

XVIII. Correction for Differences of Temperature. 392 

XIX. Correction for Differences of Gravity at Various Latitudes 393 

XX. Correction for Decrease of Gravity on a Vertical. 394 

XXI. Correction for the Height of the Lower Station. Positive 394 

XXII. Altitude by Boiling-point of Water. 403 

XXIII. Coordinates for Projection of Maps. 419 

XXIV. Lengths of Degrees of Meridian and Parallel. 437 

XXV. Arcs of the Parallel.;. 438 

XXVI. Meridional Arcs. Coordinates of Curvature.139 


xxi 


























XXII 


TABLES . 


TABLE PAGE 

XXVII. Areas of Quadrilaterals of Earth’s Surface. 445 

XXVIII. Scale Equivalents for Various Ratios. 447 

XXIX. Ratios Equivalent to Inches to One Mile. 448 

XXX. Convergence of Meridians. 540 

XXXI. Difference in Height between the Apparent and True 

Level .. 549 

XXXII. Sizes of Heliotrope Mirrors. 569 

. XXXIII. Solution of Oblique Plane Triangles. 597 

XXXIV. Pierce’s Criterion. 606 

XXXV. Factors for Computing Probable Error. 608 

- i * ■ •- ct 

XXXVI. Log m for Determining Spherical Excess. 622 

XXXVII. Factors for Computation of Geodetic Latitudes, Longi¬ 
tudes, and Azimuths. 649 

XXXVIII. Corrections to Longitude for Difference in Arc and Sine 670 

XXXIX. Values of Log -. 671 

cos l /2d 

XL. Log F ... 671 

XLI. Interconversion of English Linear Measures. 673 

XLII. Interconversion of English Square Measures. 674 

XLIII. To Convert Metric to English Measures... 675 

XLIV. To Convert English to Metric Measures. 675 

XLV. To Convert Meters into Statute and Nautical Miles.676 

XLVI. Logarithmic Constants for Interconversion of Metric and 

Common Measures. 676 

XLVII. Metric to Common System, with Factors and Log¬ 
arithms . 677 

XLVIII. Miscellaneous Metric Equivalents. 677 

XLIX. Parallax of Sun (/>) for First Day of Each Month. 691 

L. Mean Refraction (Rm) . 692 

LI. Correction ( Cb ) to Mean Refraction. 693 

LII. Correction ( Cd ) to Mean Refraction... 694 

LIII. Correction ( Ca ) to Mean Refraction. 695 

LIV. Conversion of Mean Time into Sidereal Time. 696 

LV. Conversion of Sidereal Time into Mean Time. 697 

LVI. Constants for the Interconversion of Time and Arc. 698 

LVII. Conversion of Time into Arc. 699 

LVIII. Conversion of Time into Arc (continued). 700 

LIX. Conversion of Arc into Time. 701 

LX. Approximate Local Mean Astronomic Times of the Cul¬ 
minations and Elongations of Polaris for the Year 

1900 . 715 

LXI. Intermediate Times for Above... 716 

LXII. Azimuths of Polaris at Elongation. 717 

LXIII. Corrections to Azimuths of Polaris for Each Month... 717 






































TABLES. 


XXlll 


TABLE PAGE. 

LXIV. Azimuths of Polaris. 718 

LXVI. Values of m for Every 5 0 Declination. 737 

LXVII. Reduction of Observations on Close Circumpolar Stars.. 738 

LXVIII. Correction for Differential Refraction. 739 

2.sin 2 V>r 

LXIX. Values of- —— . 740 

sin 1 

LXX. Factors for Reduction of Transit Observations. 758 

LXXI. Ration List. 834 

LXXII. Relative Times of Exposure for Different Stops and 

Subjects . 874 



























































■ 








. 

■ 





















































































LIST OF ILLUSTRATIONS. 


PLATES PAGE 

I. Surveying Under Difficulties. Frontispiece " 

II. Contour Survey of Site for Dam, Snake River, Idaho.. 58 

III. Sand Hills, Bench, Creek, etc., above Albany, N. Y. 136' 

IV. Isogonic Chart of United States for 1905. 224 


FI CURE 

1. Diagram of Plane-table Triangulation, Frostburg, Md. 

2. Roads, Houses, and Locations Resulting from Traverse, Frost¬ 

burg, Md.. 

3. Adjusted Sketch Sheet, Frostburg, Md.. 

4. Completed Topographic Map, Frostburg, Md. 

5. Land Survey Control for Topographic Sketching, North Dakota 

6. Topographic Map on Land Survey Control, Fargo, N. D. 

7. Optical Illusion as to Relative Heights of Divides. 

8. Contour Topographic Survey for Location of Mexican Central 

Railway . 

9. Detailed Contour Survey for Canal Location. 

10. Preliminary Map of Canal, Montana. 

11. Contour Survey of a Reservoir Site, Montana... 

12. Portion of Jerome Park Reservoir Survey, New York. 

13. Plan and Profile of Twin Lakes Dam Site, Colorado. 

14. Topographic and Cadastral Map of Baltimore, Md. 

15. Field-sketch Map Made on Plane-table in Alaska. 

16. Geographic Contour Map made from Fig. 15. 

17. Captain Zebulon Pike’s Map about Pike’s Peak. Colo., 1807. 

18. Captain J. C. Fremont’s Map about Pike’s Peak, 1845.. 

19. Wheeler Map about Pike’s Peak, Colo., 1876. 

20. Hayden Map about Pike’s Peak, Colo., 1875. 

21. U. S. Geological Survey Sheet about Pike’s Peak, Colo., 1892.. 

22. U. S. Geological Survey Map about Cripple Creek, near Pike’s 

Peak, 1894... 



26 

27 


3i 

38 

4 i 

45 


50 

54 

56 

59 

60 

61 
66 

71 

72 

78 

79 

80 

81 
83 


85 


XXV 



























xxvi LIST OF ILLUSTRATIONS. 

FIGURE PAGE 

23. Field Plane-table Sheet, Exploratory Route Survey, Alaska- 88 

24. Exploratory Route Survey, Alaska. Final Drawing. 89 

25. Exploratory Survey, Seriland, Sonora, Mexico. 90 

26. Skeleton of Route from Best Available Map. 94 

27. Sketch Route of Fig. 26, Filled in with Field Notes. 96 

28. Sketch Route of Fig. 26, Filled out from Field Notes of Fig. 27.. 97 

29. Reconnaissance on Nile River from Gordon’s Steamer. 98 

30. Reconnaissance Sketch of Arab Position at Abu Klea. 99 

31. Military Map of Operations about South Mountain. 101 

32. Military Siege Map. 102 

33. Cadastral Map of U. S. Public Land Survey, Indian Territory.. 106 

34. Canyon in Homogeneous Rock, Yosemite Park, Cal. 115 

35. Watergaps and Pirating Streams, Pottsville, Pa. 117 

36. Erosion in Soft Rock. 120 

37. Erosion in Hard Rock. 120 

38. Erosion in Horizontal Beds of Hard and Soft Rock. 120 

39. Erosion in Alternate Beds of Soft and Hard Rock. 120 

40. Erosion in Soft Rock Underlain by Hard. 120 

41. Volcanic Mountain, Mt. Shasta, Cal. 123 

42. Alluvial Ridge and Flood Plain, Lower Mississippi River. 125 

43. Sand Dunes, Coos Bay, Ore. 127 

44. Dissected Plateau, Northern Arizona. 129 

45. Mountain Range and Amphitheater, Irwin, Colo. 131 

46. Drawing Radial Sight Lines. 148 

47. Intersecting on Radial Lines. 149 

48. Coast Survey Plane-table. 154 

49. Telescopic Alidade and Johnson Plane-table. 155 

50. Johnson Plane-table Movement. 156 

51. Telescopic Alidade. 158 

52. Gannett Traverse Plane-table and Sight Alidade. 160 

53. Vertical Angle Sight Alidade. 161 

54. Folding Exploratory Plane-table and Small Theodolite. 163 

55. Cavalry Sketchboard and Straight-edge. 165 

55a. Batson Sketching-case. iboo 

56. Scales of the Slide Rule. 169 

57. Double Screw to Hold Plane-table Paper. 177 

58. Intersections with Plane-table. 183 

59. Three-point Locations. 186 

60. Bessel’s Graphic Solution of Three-point Problem. 189 

61. Ranging-in .. 192 

62. Lining-in . 193 

63. Two-point Problem. 194 

64. Traversing with Plane-table. 198 

65. Section of Engineers’ Transit. 203 

66. Collimation Adjustment. 206 

67. Traversing with Transit. 208 














































LIST OF ILLUSTRATIONS . XXVii 

FIGURE PAGE 

68. Plat of Transit Road Traverse. 210 

69. Full-circle Vernier Protractor. 211 

70. Three-arm Protractor. 211 

71. Signs of Latitudes and Departures. 213 

72. Prismatic Compass. 214 

73. Douglas Odometer Attached to Wheel. 230 

74. Bell Odometer. 231 

75. Hand Recorder. 232 

76. Stadia Measurement on Horizontal. 244 

77. Stadia Measurement on Slope. 248 

78. Stadia Reduction Diagram. 259 

79. Stadia Reduction Diagram to Horizontal Distances. 265 

80. Diagram for Reducing Inclined Stadia Distances to Horizontal.. 266 

•81. Speaking Stadia and Level-rods. 270 

82. Angular Tachymetry. 273 

83. Wagner-Fennel Theodolite Tachymeter. 281 

84. Reconnaissance Sketch-map with Cavalry-board and Range¬ 

finder . 285 

85. Range-finding with a Direction-point, D . 287 

86. Range-finding without Direction-point. 287 

87. Measuring Long Base with Range-finder. 287 

88. Weldon Range-finder. 288 

89. Range-finding with Plane-table. 290 

90. Photograph by Canadian Survey and used in Map Con¬ 

struction . 293 

91. Bridges-Lee Photo-theodolite. 297 

92. Projection of Camera-plates from a Station..301 

93. Projection of Photograph. 302 

94. Construction of Map from Four Photographic Stations. 303 

95. Engineer’s Wye Level. 309 

96. Target-rods . 3 12 

97. Speaking Level-rods. 3 M 

98. Turning-points . 3 1 5 

99. Illustrated Description of Bench-mark. 316 

100. Bronze Tablet and Wrought-iron Bench-mark Post.318 

ipoa. New Coast Survey Binocular Precise Level. 326 

1006. Section of Binocular Precise Level . 327 

101. Precise Spirit-level. 328 

104. Single-rodding with Two Rodmen. 332 

105. Duplicate Rodding, Both Lines Direct Only. 332 

106. U. S. Geological Survey Double-target Level-rod. 334 

107. U. S. Geological Survey Precise Speaking-rod. 336 

108. Level Circuit. 346 

109. Group of Connected Level Circuits. 347 











































xxvill LIST OF ILLUSTRATIONS. 

FIGURE PAGE 

no. Long-distance Leveling Across Tennessee River. 354 

hi. Locke Hand-level. 356 

112. Abney Clinometer Hand-level. 357 

113. Section Through Cistern and Tube of Mercurial Barometer.... 376 

114. Aneroid Barometer. 398 

115. Gnomonic and Orthographic Projections..... 407 

116. Stereographic and External Projections. 408 

117. Orthographic Equatorial Projection. 408 

118. Orthographic Horizontal Projection. 408 

119. Stereographic Equatorial Projection. 409 

120. Stereographic Meridional Projection. 409 

121. Stereographic Horizontal Projection. 409 

122. Lambert’s Surface-true Central Projection . 409 

123. Cylinder Projections. 411 

124. Equidistant Flat Projection... 41 x 

125. Mercator’s Cylinder Projection .. 411 

126. Van der Grinten’s Circular Projection. 412 

127. Babinet's Homalographic Projection of the Whole Sphere .... 413 

128. Tangent Cone Projection. 414 

129. Intersecting Cone Projection. 414 

130. Equal-spaced Conical Projection. 415 

131. Mercator’s Conical Projection.415. 

132. Equivalent Conical Projection.415 

133. Bonne’s Projection. 416 

134. Construction of Polyconic Projection. 417 

135. Contour ( D ), Shade-line ( B ), and Hachure Construction (A)... 451 

136. Shaded Contour Map. 453 

137. Sketch Contours. Xalapa, Mexico. 455 

138. Relief by Crayon-shading. 457 

139. Contour Sketch. 459 

140. Contour Projection. 461 

141. Hachure Construction. 462 

142. Shaded Hachures. 463 

143. Hachured and Contoured Hill on Different Scales. 465 

144. Conventional Signs; Public and Private Culture. 467 

145. Conventional Signs; Miscellaneous Symbols and Boundary Lines 469 

146. Conventional Signs; Hydrography. 471 

147. Conventional Signs; Relief or Hypsography. 473 

148. Conventional Signs; Lettering. 475 

149. Pantograph . 478 

150. Relief Map from Catskill Model. 481 

151. Relief Map from Contour Model. 487 

152. Coast Survey Tape-stretcher. 502 

153. Tape-stretcher for Use on Railroads. 504 

154. Simple Tape-stretcher. 505 

155. Contact-slide Base Apparatus. 509 












































LIST OF ILLUSTRATIONS. xxix 

FIGURE PAGE 

156. Eimbeck Duplex Base Apparatus. 512 

157. Cross-section of Iced-bar Apparatus. 514 

158. Repsold Base Apparatus. 515 

159. Transfer of Measured Base 00 ' to Computed and Marked 

Base AB . 525 

160. Intervisibility of Objects. 551 

161. Base Expansion. 555 

162. Eight-inch Direction Theodolite. 557 

163. Section of Micrometer through Screw Showing Comb and Cross¬ 

hairs in Central Plan. 558 

164. Quadripod Signal. 563 

165. Observing Scaffold and Signal. 567 

166. Telescopic Heliotrope. 570 

167. Steinheil Heliotrope. 572 

168. Trigonometric Functions. 594 

169. Graphic Statement of Formulas for Solution of Right-angled 

Triangles . 595 

170. Solution of Triangle Given Two Sides and Included Angle. 598 

171. Solution of Triangle Given Certain Functions. 598 

172. Solution of Triangle Given Three Sides. 599 

173. Three-point Problem. 600 

174. Reduction to Center. 609 

175. Station Adjustment. 613 

176. Angle and Side Notation. 618 

177. Angle and Side Equations. 624 

178. Computation of Azimuths. 642 

179. Latitude. Declination, and Altitude. 683 

180. Conversion of Time . 698 

180a. Aspects of Polaris... 7 1 3 

181. Astronomic Transit and Zenith Telescope. 727 

182. Chronograph . 75 ° 

183. Sextant . 778 

184. Graphic Illustration of the Solar Attachment. 783 

185. Smith Meridian Attachment. 786 

1 5a. Baldwin Solar Alidade. 79 2 

186. Where a Pack-mule Can Go. 813 

187. A Pretty Camp Ground, North Carolina. 815 

188. Wall-tent with Fly. 818 

189. Ridge and Pole for Wall-tent. 822 

190. Shelter Tents. 823 

191. Jointed Shelter-tent Poles. 824 

192. Sod-cloth and Ditch. 826 

193. Tent Stove and Pipe. 827 

194. Folding Camp Table. 829 

195. Folding Tin Reflecting Baker. 832 

196. Camp Wagon. 837 

197. Full-rigged Moore Army Pack-saddle. 840 
















































XXX LIST OF ILLUSTRATIONS. 

FIGURE PAGE 

198. Pack-saddle Cinches. 840 

199. Lashing Pack with Diamond Hitch. 842 

200. Loading Pack-mule with Mess-boxes. 843 

201. Packing on Men’s Backs, Adirondacks. 846 

202. Pack-basket . 847 

203. Plane-table Station on Mountain in Alaska. 851 

204. Inducing Artificial Respiration. 859 

205. Diagram Showing Relative Exposures at Different Times of Day 

and Year. 873 










PART I. 


TOPOGRAPHIC , GEOGRAPHIC , EXPLORATORY 

SURVEYING. 


CHAPTER I. 

KINDS OF MAP SURVEYS. 

I. Classes of Surveys —Surveys may be grouped under 
three general heads: 

1. Those made for general purposes, or information sur¬ 
veys. 

2. Those made for jurisdictional purposes, or cadastral 
surveys. 

3. Those made for construction purposes, or engineering 
surveys. 

Information surveys may be exploratory, geodetic, geo¬ 
graphic, topographic, geologic, military, agricultural, mag¬ 
netic, or hydrographic. Geodetic surveys are executed for the 
purpose of determining the form and size of the earth. 
They do not necessarily cover the entire surface of the 
country, but only connect points distant from each other 
20 to 100 miles. Topographic and geographic surveys are 
made for military, industrial, and scientific purposes. To 
be of value they must be based upon trigonometric or tri¬ 
angulation surveys, but not necessarily of geodetic accuracy. 

The mother map, or that from which all others are derived, 
is the topographic map. This is made from nature in the field 



2 


KINDS OF MAP SURVEYS. 


by measures and sketches on the ground. It is the original or 
base map from which can be constructed any variety of maps 
for the serving of separate purposes. The historian may 
desire to make a map which will indicate the places upon 
which were fought great battles, or on which are located the 
ancestral estates of historic families. The geologist may 
desire to indicate the location of certain rock formations. 
The promoter of railways or other engineering works may 
desire to represent the route of his projected road or the 
location of city water-supplies or real-estate subdivisions. For 
these several purposes the topographic or base map furnishes 
the original data, or foundation, on which can be indicated, in 
colors or otherwise, any special class of information. 

Cadastral surveys define political and private property 
boundaries and determine the enclosed areas. .Such surveys 
are executed for fiscal and for proprietary purposes, and their 
value depends upon the degree of accuracy with which they 
are made. A cadastral survey is not necessarily based upon 
triangulation and may be only crudely executed with com¬ 
pass and chain. To thoroughly serve its purpose, however, 
it should be based on geodetic work of the greatest refine¬ 
ment. It does not necessarily cover the entire area en¬ 
closed, but only points and lines which mark the boundaries. 

Engineering surveys are executed in greater detail than 
any of the above. They may preferably follow some of 
them and are preliminary to the construction of engineer¬ 
ing works. They are conducted with such detail as to per¬ 
mit the computing of quantities of materials to be moved 
and the exact location of the various elements of the works 
which are to be constructed. Engineering surveys may be 
made for the construction and improvement of military 
works, as forts, navy yards, etc. ; for constructing routes of 
communication, as roads, electric lines, canals; for reclama¬ 
tion of land, as irrigation and swamp surveys; for the im¬ 
provement of natural waterways, as river and harbor sur- 


INFORMATION SURVEYS. 


3 


veys; or for the improvement of cities, as city water-supply 
and sewage disposal. 

2 . Information Surveys. — All surveys have a twofold 
purpose: 

1. To acquire certain information relative to the earth; 

and 

2 . To spread this among the people. 

The acquirement of the information is the field survey. 
The dissemination may be in the form of manuscript, illustra¬ 
tions, or sketch maps, as in the case of exploratory surveys; 
of a map only, as in the case of topographic surveys when 
the map embodies the whole result; or it may be a combina¬ 
tion of the two, as in the case of geographic surveys. 

In addition to the above primary classes of information 
surveys are the numerous minor differences in the method of 
field-work, including the instruments used, the degree of care 
in obtaining the information, and the mode of recording the 
results in notes or maps. The instrumental work of explora¬ 
tory surveys is usually of the crudest and most haphazard 
kind, the observations having to be taken and the notes re¬ 
corded incidentally and by such means and at such time as 
the primary necessities of the expedition, those of moving 
forward over the route traversed, will permit. Moreover, 
from the necessity of the circumstances such surveys are 
rarely homogeneous, never covering completely any given 
area; else they would cease to be exploratory. Being dis¬ 
connected, they are fixed from time to time with relation to 
the earth by such astronomic observations as will fre¬ 
quently check the interrupted route surveys in relation one 
to the other. 

Topographic and geographic surveys differ essentially 
from exploratory surveys, but from each other only in minor 
details of scale, degree of representation of relief, and the 
note taken of the sphericity of the earth. Topographic sur¬ 
veys are generally executed on so large a scale and with such 


4 


KINDS OF MAP SURVEYS. 


care and detail that account need rarely be taken of the 
sphericity of the earth in plotting the resulting map, and they 
are therefore based on geodetic data only as they merge into 
geographic surveys. Moreover, all important natural and 
artificial features may be represented on the resulting map 
because of its large scale. 

Geographic surveys merge imperceptibly, on the one hand, 
into topographic surveys, as the scale of the latter becomes 
so small and the area depicted on a given map sheet so large 
that the shape of the earth must be considered. On the 
other hand, they may be plotted on so small a scale and the 
relief be depicted by such approximate methods that they 
merge imperceptibly into exploratory surveys, being practi¬ 
cally of the same nature as the latter excepting that they 
cover a given area in its entirety. 

3. Topographic Surveys. — A topographic map is one 
which shows with practical accuracy all the drainage, culture, 
and relief features which the scale of representation will per¬ 
mit. Such scale may be so large and the area represented 
on a given map sheet be so small that the control for the 
field surveys will be procured by means of plane and not of 
geodetic surveying. On the other hand, the scale may be so 
small and the area represented on the given map sheet so 
large as to require control by geodetic methods. 

The mistake is often made of assuming that a topographic 
map is special and not general. It is general, as it is not 
made for the purpose of constructing roads and highways, 
though it becomes a very valuable aid in their projection; 
nor is it made for the purposes of reclaiming swamp-land or 
irrigating arid land, but it furnishes general information 
essential to a preliminary study and plan for their improve¬ 
ment. The outcome of a topographic survey being a topo¬ 
graphic map, it should be judged by the map, and the map 
should be judged by the manner in which it serves the gen¬ 
eral purpose. Above all, of two maps or works of any kind 


FEATURES ON TOPOGRAPHIC MAPS. 5 

made for the same purpose and serving that purpose equally 
well, that the cheaper one is the better is a well-recognized 
principle of engineering. 

In the prosecution of a general topographic survey only 
such primary points should be determined geodetically as are 
essential to the making of the map. About one such point 
per one hundred square miles is a fair average for a one-mile to 
one-inch map. Such spirit-level bench-marks should be set 
and recorded as are obtained in carrying bases for levels over 
the area under survey. On the above scale about one bench 
to five square miles is a fair average. 

4. Features Shown on Topographic Maps. —The fea¬ 
tures exhibited on topographic maps may be conveniently 
grouped under the three following heads: 

1. The hydrography, or water features, as ponds, streams, 
lakes. 

2. The hypsography, or relief of surface forms, as hills, 
valleys, plains. 

3. The culture, or features constructed by man, as cities, 
roads, villages, and the names printed upon the map. 

In order that these various features may be readily dis¬ 
tinguishable and thus give legibility to the map, it is usual to 
represent the hydrography in blue, the relief in brown, and 
the culture in black. In addition to this, wooded areas may 
be indicated in a green tint. 

The object of a topographic survey is the production of a 
topographic map. Hence the aim of the survey should be to 
produce only the map; neither time nor money should be 
wastefully expended in the erection or refined location of 
monuments; the demarkation of public or private boundary 
lines; or the establishment of bench-marks beyond such as 
are incidental to the work of obtaining field data from which to 
make the map. The erection, location, and description of 
boundary marks is the special work of a property or cadastral 
survey. The erection, description, and determination of 


6 


KINDS OF MAP SURVEYS. 


monuments and bench-marks as primary reference points is 
the work of a geodetic survey. The determination of many 
unmarked stations for map-making purposes is the work of a 
topographic survey. 

5. Public Uses of Topographic Maps. —For the purposes 
of the Government or State good topographic maps are in¬ 
valuable. They furnish the data from which the congressman 
or the legislator can intelligently discover most of the informa¬ 
tion bearing directly upon the problem in hand, and they give 
committees great assistance in their decisions as to the need 
of legislation. If a River and Harbor bill is before Congress, 
or a bill relating to State Canals before the Legislature, by an 
inspection of such maps the slopes of the country through 
which the canal is to pass or in which the improvements are 
to be made may be readily ascertained. The sources of water- 
supply for a canal or river may be accurately measured on 
such a map and their relation to the work in hand intelligently 
ascertained. 

If the War Department of the Government or the Adjutant- 
General’s Office of the State desires to locate an arsenal, en¬ 
campment ground, or other military work, or, above all, if it 
is to conduct active military operations in the field, such maps 
serve all the preliminary purposes of the best military maps. 
With the addition of a very little field-work during war times, 
such as the indication of ditches, fence lines, outbuildings, 
etc., on the mother or topographic map, a perfect military 
map may be obtained. 

For the Post-office Department or private stage, express, or 
telephone companies, such maps furnish the basis on which 
an accurate understanding can be had of contracts submitted 
for star or other routes for carrying the mails or packages. 
As these maps show the undulations of the surfaces over 
which roads pass, their bends and the relative differences in 
length, the difficulties in travel on competing roads can be 
readily ascertained from them. 


PUBLIC USES OF TOPOGRAPHIC MAPS. 


7 


The Land Departments of the Government and State can 
discover on such maps not only the outlines of the property 
under their jurisdiction, but its surface formation. Forestry 
Boards can see indicated upon these maps the outlines of the 
various wooded areas, besides the slopes of the lands on which 
these woods are situated, their relation to highways of trans¬ 
portation, railways, or streams, and the slopes to be encoun¬ 
tered in passing through the woods on these highways. 

The Legal department of the Government or State finds 
these maps of service in discussing political or property bound¬ 
ary lines, in ascertaining within what political division crimes 
are committed, or individuals reside with whom the officers of 
the law desire to communicate. It is difficult to see how any 
systematic or economical plan of road improvement can be 
advantageously made without the knowledge of existing grades, 
the physiography of the district through which the roads pass, 
and the location of quarries, which such maps present. 

The whole system of making successive special surveys or 
maps for every new need is one of the most wasteful in our 
present public practice, nor can it be otherwise until one survey 
shall be made that answers all important official uses. The 
amount of money that has been expended in making small 
maps of numerous cities and villages would have mapped, on 
a general scale, many times the area of the country. Even 
when we have these special maps they do not fully answer the 
purpose for which they were intended, as they only show the 
small area included within the immediate plan of operations. 
The value of a stream for economic purposes cannot be fully 
ascertained bv an examination of the stream at the point from 
which it is to be used, but the drainage basin from which it 
derives its supply should be surveyed, and its area and slopes 
be known. A good topographic map not only shows the re¬ 
lations between the natural and artificial features in the im¬ 
mediate neighborhood under consideration, but it shows the 
relations of these to the surrounding country. 


8 


KINDS OF MAP SURVEYS. 


6 . Degree of Accuracy Desirable in Topographic Sur¬ 
veys. —It is difficult to set any standard for the amount of de¬ 
tail which the topographer must sketch on his map, or the 
amount of control which must be obtained for the checking 
of this detail. A topographic map may be so made as to serve 
many useful purposes and yet be almost wholly a sketch, 
scarcely controlled by mathematical locations. The same 
territory may be mapped on the same scale with little improve¬ 
ment in the quality of representation of topographic form and 
yet the work be done with such detail and accuracy and such 
amount of control as to make it useful for all practical pur¬ 
poses to which its scale adapts it. 

With these facts clearly in view, it is evident that explicit 
instructions to the topographer are a practical necessity. Un¬ 
like any other surveyor the topographer must use his own 
judgment or be guided by instructions regarding the amount 
of time and money to be spent in obtaining detail and control, 
since the latitude permissible in mapping the same territory 
on the same scale varies greatly according to the uses to 
which the map is to be put. Such instructions should in¬ 
terpret the significance of scale and contour interval, and 
should cover the technical details of operations as found ap¬ 
plicable to conditions and locality (Art. 7). They should 
also fix the method of making and preserving field-notes. 
There are a variety of methods of survey, of instruments, and 
of records which are generally applicable to any case, yet to 
the expert topographer there is practically only one best way 
for each, and this can be decided only after he has inspected 
the country or has otherwise acquired knowledge of its 
characteristics. 

The scale and mode of expressing relief (Art. 191) must 
be fixed as well as the contour interval, if contours are em¬ 
ployed, in order that all the data necessary for the construc¬ 
tion of the map on this scale may be obtained. The methods 
and instruments should be stated in order that those best 


A CCURA CY D ES1RA BLE. 


9 


suited to the conditions may be selected in the beginning-. 
The mode of record should be fixed in order that there may 
be uniformity in the results brought into the office, provided 
there are various topographers working on the same area. 
Such instructions are to the topographer what specifications 
are to the contractor, yet they cannot quite carry the force 
of law because of the unforeseen exigencies which may arise 
and which require departure from fixed instructions in ac¬ 
cordance with the best judgment of the topographer. 

In topographic mapping it is sometimes desirable to make 
hasty preliminary or reconnaissance maps of a region in order 
that some information of the area may be immediately ob¬ 
tained. Such maps are practically sketches covering an ex¬ 
tensive area and without adequate framework of control, yet 
they contain most of the information required in the early de¬ 
velopment of the region. The error has too frequently been 
made of giving such maps the ear-marks of accuracy by rep¬ 
resenting the relief by numbered contours. In this they are 
misleading. Contours indicate precision and should justly be 
taken as accurate within the limits of the map scale. As has 
been aptly stated by Mr. J. L. Van Ornum, “ accuracy is ex¬ 
pected where exactitude is shown, and the conclusion is just 
that inaccuracy in representation is inexcusable.” Where 
for any reason the desired accuracy cannot be attained for 
lack of the proper control, the resulting map is merely a 
sketch-map, and relief should be indicated not by contours but 
by hachures or by sketched contours; that is, lines in contour 
form, but disconnected and unnumbered. Such sketch-maps 
are useful as representations of topographic form, but are 
valueless as base-maps on which to plan great public improve¬ 
ments, the inception of which is so closely connected with 
topographic surveys. 

A topographic map well executed is, to quote Captain 
George M. Wheeler, “the indispensable, all-important sur¬ 
vey, being general and not special in character, which under- 


IO 


KINDS OF MAP SURVEYS. 


lies every other, including also the graphic basis of the eco¬ 
nomic and scientific examination of the country. This has 
been the main or principal general survey in all civilized coun¬ 
tries. The results of such a survey become the mother source 

*! 

or map whence all other fiscal examinations may draw their 
graphic sustenance.” Such a characterization of a topographic 
survey can apply only to one accurately made and on which 
every feature represented is as accurately shown as the scale of 
map warrants. 

In planning a topographic survey the controlling factor of 
the scale must always be kept clearly in mind, as this is the ul¬ 
timate criterion which decides the method of survey and the 
amount of time and money to be expended in its execution. 
The underlying law of topographic mapping is that applied to 
other engineering works, namely, no part of the construction, 
nor any part of the survey, should be executed with greater 
detail or at greater expense than will permit it to safely per¬ 
form the duties for which it is intended. Thus, in mapping 
an extended area, traverse methods alone for horizojital con¬ 
trol are insufficient unless performed with the greatest exacti¬ 
tude. The primary triangulation on which such a survey is 
based should be no more accurate than will permit of plotting 
the points with such precision that they shall not be in error 
by a hair’s breadth at the extreme limit to which the triangu¬ 
lation is extended. The secondary triangulation should be 
executed with only such care as will permit of plotting with¬ 
out perceptible error on the scale selected and within the lim¬ 
its controlled by the nearest primary triangulation points. Sim¬ 
pler methods of securing horizontal control may be adopted 
for the minor points within the secondary triangulation, and 
these methods, be they by plane-table triangulation (Chap. 
IX) or by traverse (Chap. X), need be nothing better than 
will assure the plotting of the result without perceptible error 
and within distances controlled by the nearest secondary tri¬ 
angulation points. Finally, minor details maybe obtained by 


AMOUNT OF CONTROL REQUIRED. II 

the crudest methods of traverse, range-finding, pacing, or 
sketch-board (Arts. 81, 116, 95, and 61), providing that the 
distances on the map over which such methods are propagated 
shall be so small as to warrant their not being perceptibly 
in error within the limits of the controlling points f the next 
higher order. 

/ 

As with the horizontal control so with the vertical control , 
no more time should be expended or precision attempted 
in determining elevations than are necessary to obtain the 
data essential to the mapping of the relief accurately to the 
scale limit. Where relief is to be represented by contours of 
a small interval and on a large scale, or where the slopes of 
the country are gently undulating or comparatively level, the 
leveling must be of a high order that the contours may be ac¬ 
curately placed in plan. In country having slopes as gentle 
as 5 to 10 feet to the mile, a difference of a few feet in 
elevation may mean that distance in error in the horizontal 
location of the contour if the elevations are not determined 
with accuracy. On the other hand, in precipitous mountain 
country much less care is necessary in the quality of the level¬ 
ing, since a large error in vertical elevation may be represented 
in plotting by the merest fraction in horizontal plan. For a 
large contour interval in country of moderate slopes less ac¬ 
curacy is essential in the determination of the elevation. 
For contours of 20 feet interval errors of elevation varying 
from 5 to 20 feet or more may be made, depending upon the 
steepness of the slope and the consequent nearness in hori¬ 
zontal plan of one contour to the next. The same ratio 
applies to greater contour intervals. Therefore the methods 
pursued in determining the elevations should begin with a 
careful framework of spirit-leveling (Art. 129), and the 
amount of this should be only so great as to insure that the 
dependent levels of less accuracy shall not be so far in error 
as to be appreciable for the scale and contour interval selected 
and for a given slope of country. Based on these spirit-levels 


12 


KINDS OF MAP SURVEYS . 


rougher elevations by vertical angulation with stadia (Art. 
102) or by trigonometric methods (Art. 159) may be employed, 
and tied in between these may be elevations by aneroid (Art. 
174), the latter being checked at intervals sufficiently frequent 
to assure that the resulting elevations shall not introduce ap¬ 
preciable errors in the location of contours. 

The same rules should apply to the frequency with which 
vertical control points are determined. These should be so 
close together for the scale of the map and for the contour 
interval selected that in connecting them by eye in the 
course of the sketching no error appreciable on the scale 
shall be introduced. Any map, the best obtainable, is but a 
sketch controlled by locations. No one would undertake to 
determine the elevation and horizontal plan of every point on 
a contour line. Control positions on contours are only deter¬ 
mined with sufficient frequency to insure comparative accu¬ 
racy in connecting them. Bearing on this same point is the 
fact that such connection by sketching can undoubtedly be 
done with greater accuracy on the plane-table board with the 
terrane in view than from notes platted up in office or from 
photographs or profile drawings. 

Where relief is to be represented by hachures or broken 
sketch contours, precision in absolutely fixing the vertical ele¬ 
ment is of the least moment. It is generally desirable in 
making such maps to write approximate altitudes at promi¬ 
nent points, as stream junctions, villages, or mountain sum¬ 
mits, but the chief desideratum is relative differences in eleva¬ 
tion in order that the number of the sketched contours and 
their frequency, or the degree of density of the hachuring, 
may give an index to the amount of relief. 

7. Instructions Relative to Topographic Field-work. 
—The following instructions are those issued by the Director 
of the United States Geological Survey for the guidance of 
topographers in the field : 

I. At least two primary triangulation points or a primary control line 


INSTRUCTIONS TO TOPOGRAPHERS. 


13 

1. All primary triangulation points and primary control lines should 
be platted on each atlas sheet previous to commencing field-work. 

2. All existing map material should be diligently sought for; such of 
this as may be of value, as public-land plats, railroad, water-supply, city, 
Coast Survey, Army Engineer, or other public or private material, should be 
carefully compiled. If on field inspection this proves adequate, it should be 
brought up to date and incorporated in the field sheets. 

3. On each atlas sheet, in addition to primary levels, such other eleva¬ 
tions should be obtained instrumentally that aneroids when used need never 
be left without a check elevation for distances exceeding 2 \ to 3 miles. These 
control elevations may come from profiles of railroads, from spirit-levels 
or from vertical angulation. 

4. Plane-table triangulation must be conducted on the large sheets, 
and it is desired that as fast as intersections are obtained by the topog¬ 
rapher the vertical heights of stations and intersected points should be 
computed. 

5. In conducting plane-table triangulation, as many prominent features 
as possible, including hilltops, churches, and houses, should be intersected 
in order to furnish connections with the traverse work, while gaps or passes 
and salients on ridges should also have their positions and elevations deter¬ 
mined from the plane-table stations. 

6. Secondary topographic control must precede topographic sketching 
and the filling in *of minor details of the map. 

7. Field sheets must be as few in number and as large as the character 
of the topography will permit, and all main control must be adjusted thereon; 
this to be done before the filling in of minor detailed sketching is commenced. 
These minor details may be obtained by traverse on separate sheets, but 
must at once be transferred to and adjusted on main field sheets, so that no 
uncompleted spaces shall be left on them in the field. 

8. The stage of water in rivers to be shown on the topographic maps is 
to be that which exists during the greater portion of the year and represents 
the normal condition of the stream. When any other condition is represented 
an explanatory note giving stage and date should be inserted in the 
legend. 

9. The topographer in charge will be held responsible not only for the 
quality of the topographic work but also for the quality and management 
of the spirit-leveling done under his direction, and for the location and mark¬ 
ing of the bench marks, each of which he should endeavor to examine person¬ 
ally. Standard bench marks should be indicated on field sheet. 

10. Only so much of the field sheets should be inked in the field as can 
be done with sufficient care to permit of their being accepted as final draw¬ 
ings and of their being directly photographed or photolithographed (except 
where land-survey plats are used as field sheets). Accordingly, only such 


14 


KINDS OF MAP SURVEYS . 


inks should be used as will photograph readily—mixed burnt sienna for 
contours, black for culture, and mixed Prussian blue for drainage. 

11. A full record must be made on the title-page of each notebook, stating 
character of work, locality, atlas sheet, and date of record; also name of 
topographer and maker of notes. 

12. Plats, on a large scale, should be made or obtained of all large 
cities, showing the streets and houses in detail. 

13. The determination and spelling of names of streams, mountain peaks, 
villages, and other places of note should receive particular attention. 

14. Plane-table stations must be numbered consecutively with Roman 
numerals. If the station has been sighted before occupation, the station 
number must be followed by the number of the sight to it. Sights or points 
must be numbered by Arabic numerals consecutively; and a point once 
numbered must always be given the same number whenever recognized. 
If the points sighted exceed one thousand in number, a capital M shall be 
written at the head of the number column of the notebook. This rule must 
be followed by all members of the topographic branch. 

15. The standard conventional signs must be used on all plane-table and 
traverse sheets. 

16. The sheets must be inked clearly and carefully, with uniformity 
throughout, and in such manner as to adapt them for one-third reduction to 
publication scale. Only such ink should be used as will photograph readily. 
(See par. 10, above.) Culture should be inked first, and standard conven¬ 
tional signs used. 

17. In drawing streams care should be taken that the lines shall not 
become faint and uncertain near the sources of the streams, and the placing 
of drainage in every little gully simply to indicate that it may be a water¬ 
course should be avoided. Use the symbol dash and three dots for intermittent 
streams. 

18. In lettering, names parallel to the east or west sides of the sheet should 
read from the south side. Names of minor importance and figures of eleva¬ 
tion should be placed close to the object, on the right and horizontally. The 
letters, figures, and cross pertaining to a bench mark should be arranged 
with the letters above and to the left of the cross, and the figures below and 
to the right. 

19. The original drawing of a topographic sheet shall be verified by 
some competent person in addition to the topographer who compiles it, 
by comparison with field sheets, and such “proof-reading” shall be recorded 
on the appropriate form. 

8. Elements of a Topographic Survey. —From a con¬ 
structive point of view a map is a sketch corrected by loca¬ 
tions. The making of locations is geometric, that of sketch- 


ELEMENTS OF A TO POOR A PHIC SURVEY. I 5 

ing is artistic. However numerous may be the locations they 
form no part of the map itself, serving merely to correct the 
sketch which supplies the material of the map. Every map, 
whatever its scale, is a reduction from nature and conse¬ 
quently must be more or less generalized. It is therefore im¬ 
possible that any map can be an accurate, faithful picture of 
the country it represents. The smaller the scale the greater 
the degree of generalization and the farther must the map de¬ 
part from the original. The larger the scale the smaller the 
area brought together on a given map, and the less it appeals 
to the eye which grasps so extended a view of nature. There 
is, however, for the purposes of making information maps, a 
scale which is best suited to every class of topography, and 
the best result only will be obtained by selecting the relation 
of horizontal scale and contour interval which fits the partic¬ 
ular topography mapped. 

By far the most important work of topographic mapping 
is the sketching (Arts. 13, 15, 1 7, and 193), and this should 
be done by the most competent man in the party—presum¬ 
ably its chief. He should not only sketch the topography 
because of his superior qualifications for that work, but also 
because the party chief is responsible for the quality of all the 
work, and only in the sketching, which is the last act in map¬ 
making, has he full opportunity for examining the quality of 
the control and of the other elements of the work executed 
by his subordinates. The map-sketcher is therefore the topog¬ 
rapher, and it is in the matter of generalization or in the se¬ 
lection of scale and the amount of detail which should be 
shown for the scale selected that the judgment of the topog¬ 
rapher is most severely tested. This is the work in which the 
greatest degree of proficiency can only be attained after years 
of experience. The topographer must be able to take a broad 
as well as a detailed view of the country, and to understand 
the meaning of its broadest features that he may be able best 
to interpret details in the light of those features (Chap. VI). 


16 


KINDS OF MAP SURVEYS. 


It is only thus that he can make correct generalizations, and 
thus that he is enabled to decide which detail should be 
omitted and which preserved in order to bring out the pre¬ 
dominant topographic features of the region mapped. 

The correctness of the map depends upon: 

(1) The accuracy of the locations; 

(2) Their number per square inch of map; 

(3) Their distribution; 

(4) The quality of the sketching. 

The first three of these elements defines the accuracy of 
the map, and the greatest accuracy is not always desirable be¬ 
cause it is not always economical. The highest economy is 
in the proper subordination of means to ends, therefore the 
quality of the work should be only such as to insure against 
errors of sufficient magnitude to appear upon the scale of 
publication (Art. 6). The above being recognized, it is evi¬ 
dently poor economy to execute triangulation of geodetic re¬ 
finement for the control of small-scale maps, and, providing 
the errors of triangulation are not such as are cumulative, the 
maximum allowable error of location of a point on which no 
further work depends may be set at .01 of an inch on the 
scale of publication. 

The second condition, the number of locations for the 
proper control of the sketching, is not easily defined. It de¬ 
pends largely upon the character of the country and the scale 
and uses of the map. Any estimate of it must be based on 
unit of mapped surface and not of land area. For rolling or 
mountainous country of uniform slopes or large features (Fig. 
4), from i-J to 3 locations and 2 to 5 inches of traverse per 
square inch of map should, with accompanying elevations, be 
sufficient. On the other hand, in highly eroded or densely 
wooded country (Fig. 34) as many as 3 to 6 locations and 5 to 
10 inches of traverse, per inch of map may, with accompany¬ 
ing elevations, be necessary to properly control the sketching. 
Again, in very level plains country (Fig. 6) less than one lo- 


ACCURACY OF MAP. 


17 

cation and but 2 to 5 inches of traverse, with accurate eleva¬ 
tions, will suffice to furnish adequate control. 

The same is true of the third element of accuracy, the 
distribution of locations. In rolling, hilly country of uniform 
slope the control should be obtained chiefly at tops and bot¬ 
toms and changes of slope. The same is true of heavy moun¬ 
tains, excepting that all summits and gaps on ridges must be 
fixed, as well as all changes in side slopes and a few positions 
distributed about the valley bottoms. In flat plains the posi¬ 
tions determined should be locations on the contours themselves 
and at changes in their direction. In highly eroded regions 
locations of all kinds should be distributed with consider¬ 
able uniformity, so as to control every change of feature or 
slope. 

The fourth element, the quality of the sketching, depends 
wholly upon the artistic and practical skill of the topographer 
—in other words, upon his possession of the topographic 
sense, which may be described as his ability to see things in 
their proper relations and his facility in transmitting his im¬ 
pressions to paper. This is by far the most important and 
difficult requirement to meet, and one which takes a longer 
apprenticeship on the part of the topographer than all the 
others combined. 


CHAPTER II. 


SURVEYING FOR SMALL-SCALE OR GENERAL MAPS. 

9. Methods of Topographic Surveying. —Three general 
methods of making topographic surveys have usually been 
employed in the past: 

First, traversing or running out of contours by means of 
transit, chain or stadia, and level; 

Second, cross-sectioning the area under survey with the 
same instruments; and 

Third, triangulation of the territory under survey with 
such minuteness as to get a sufficient number of vertical and 
horizontal locations to permit of connecting these in office by 
contour lines. 

All three methods are slow and expensive, while the first 
two are unfitted to the survey of large areas, because of the 
inaccuracies introduced in linear or traverse surveys. 

A fourth method, and that which this book is designed to 
expound , is that always employed by the United States Geo¬ 
logical Survey as well as to a lesser degree by several other 
American and European surveys. It is fitted to make topo¬ 
graphic maps for any purpose, on any scales, and of any 
area. This consists of a combination of trigonometric, 
traverse, and hypsometric surveying to supply the controlling 
skeleton, supplemented by the “ sketching in ” of contour 
lines and details by a trained topographer. In this method 
the contour lines are never actually run out nor is the country 
actually cross-sectioned. Only sufficient trigonometric con¬ 
trol is obtained to tie the whole together, the minor control 

18 


METHODS OF TOPOGRAPHIC SURVEYING. 19 

between this being filled in : first, in the most favorable tri¬ 
angulation country almost wholly by trigonometric methods; 
second, in less favorable triangulation country by traverses . 
connecting the trigonometric points. 

There are two general methods of making a contour topo¬ 
graphic map which have been aptly styled the “regular’* 
and the “irregular.” These might be respectively called 
the old and the new. The old or regular method includes the 
surveying and leveling of a skeleton work of controlling 
traverse or triangulation and the cross-sectioning of the ter- 
rane into rectangular areas, the outlines of which are trav¬ 
ersed and leveled. In addition the leveled profiles and trav¬ 
erses are continued between this gridironing at places where 
important changes of slope occur, and finally the survey and 
leveling of flying lines or partial sections is extended from each 
station. By this method the base of each level section or the 
contour line or line of equal elevation is determined by setting 
the instrument in position where this level line intersects the 
profile, and using the telescope as a leveling instrument with 
its cross-hairs fixed on a staff at the height of the optical axis, 
a line is then located by tracing successive positions of a 
stadia rod or by locating by intersection successive points on 
the level line, and a line drawn through these points locates 
the contour curve. In addition, parts of several level sec¬ 
tions are plotted from one station by intersection on, or location 
of a staff, and by the determination of its height above or 
below the instrument by vertical angulation. In this mode 
of topographic surveying pegs are usually driven at regular 
intervals and their heights determined by spirit-level and ver¬ 
tical angulation. 

The new or irregular method of topographic surveying 
consists in determining by trigonometric methods the po¬ 
sition and height of a number of critical points of the terrane 
and connecting these by traverses and levels, not run on a cross- 
section or rectangular system, but irregularly, so as to give 


20 


SURVEYING FOR SMALL-SCALE MARS. 


plans and profiles of the higher and lower levels of the country, 
as ridge summits or divides and valley bottoms or drainage lines, 
such lines being run over the most easily traversed routes, as 
trails or roads. With the numerous positions and heights deter¬ 
mined by the triangulation, and on these traverses as controlling 
elements, contour lines are sketched in by eye and by the aid 
of the hand-level on a plane-table with the country in con¬ 
stant view. This is the method now generally employed by 
expert topographers, and the work is so conducted that the 
development of the map proceeds with the survey of the 
skeleton and rarely necessitates the return to a station when 
once occupied. Moreover, it calls for the location of less 
points and the running of fewer traverses and profiles, and 
these over more easily traveled routes, than the former 
method. It is therefore more expeditious, cheaper, and the 
resulting map is a better representation of the surface, as it 
possesses not only the mathematical elements of instrumental 
location, which in the old method are arbitrarily connected in 
office, but also the artistic element produced by connecting 
the lines of equal elevation in the field, with the country at 
all times immediately before the eye. 

io. Geological Survey Method of Topographic Survey- 
ing.—I n average country, favorable for triangulation, com¬ 
paratively clear of timber and well opened with roads, a skele¬ 
ton trigonometric survey (Chap. IX) is made, by which the 
positions and elevations of all summits are obtained, as well 
as the horizontal positions of a few points in villages or at 
road crossings, junctions, etc. This constitutes the upper 
system of control (Fig. i). Below and between this is a net¬ 
work of road traverses (Chap. X) supplemented by vertical- 
angulation (Chap. XVII) or spirit levels (Chap. XV) for 
elevations, and these follow the most easy routes of travel, 
not cross-sectioning the country in the true sense, but follow¬ 
ing all the lower lines or stream bottoms, as well as the 
gradients pursued by roads (Fig. 2). Between these two 


RATIONAL METHOD OF TOPOGRAPHIC SURVEYING . 21 


upper and lower sets of control points there are therefore 
many intermediate ones obtained by road traverses, and the 
topographer, by observation from the various positions which 
he assumes and with the knowledge he possesses of topographic 
forms, sketches the direction of the contour lines. These are 
tied accurately to their positions by the large amount of mathe¬ 
matical control already obtained, supplemented by additional 
traverses or vertical angles where such are found wanting. 
(Art. 162.) 

The instruments used are as various as are the methods of 
survey employed; the essential instruments being the 
plane-table and the telescopic alidade (Chap. VII), which 
invariably replace the transit (Art. 85) or compass (Art. 91),. 
so that all surveying is accompanied by mapping at the same 
time, and there is no tedious and confusing plotting from field- 
notes to be done later in office. Nor are any of the sa¬ 
lient features of the topography of the region lost through 
neglect to run traverses or obtain positions or elevations, all 
omissions of this kind being evident from an inspection of 
the map while in process of construction. The distances are 
obtained by triangulation with the plane-table (Art. 73) and 
by odometer measurements (Art. 98), supplemented off the 
roads by stadia measures (Chap. XII) or in very heavily 
wooded country by chaining (Art. 99) and pacing (Art. 95). 

The underlying principles of this method of topography 
are, first, a knowledge of and experience in various methods 
of surveying, and a topographic instinct or ability to appreciate 
topographic forms, which is acquired only after long practice;: 
and, second, a constant realization of the relation of scale to the 
amount of control required and methods of survey pursued ; 
no more instrumental work being done than is actually re¬ 
quired to properly control the sketching, and no more accu¬ 
rate method being employed than is necessary to plotting 
within reasonable limits of error. Thus, where trigonometric 
locations (Chap. IX) are sufficiently close together, crude 


22 


SURVEYING FOR SMALL-SCALE MARS. 


odometer traverses (Art. 98) or even paced traverses (Art. 95) 
can be run with sufficient accuracy to tie between these with 
inappreciable errors. Where trigonometric locations are more 
distantly situated, the spaces between them must be cut up 
by more accurate traverses, as those with stadia (Chap. XII) 
or chain (Art. 99), these again being gridironed by less accu¬ 
rate odometer or paced traverses. Again, a primary system 
of spirit-leveling (Chap. XV) or accurate vertical triangula¬ 
tion (Chap. XVII) is employed only for the larger skeleton, 
these elevations being connected by less accurate vertical- 
angle lines or flying spirit-levels, and these again by aneroid 
(Art. 176), each method being employed in turn so that the 
least elements of control obtained may still be plotted well 
within a reasonable limit of error in horizontal location of 
contour line. 

Finally, speed and economy are obtained by traveling 
the roads and trails in wheeled vehicles or on horseback, at a 
rapid gait from instrument station tO' instrument station; the 
slower process of walking being only resorted to where roads 
and trails are insufficient in number to' give adequate control 
and view of every feature mapped. 

II. Organization of Field Survey. —The party organiza¬ 
tion and the method of distributing the various functions of 
topographic surveying among the members of the party must 
necessarily differ with the scale of the map and the character 
of the region under survey. The work involved in making a 
topographic or geographic map may comprise four operations: 

First. The location of the map upon the surface of the 
earth by means of astronomic observations. 

Second. The horizontal location of points, which is usually 
of three grades of accuracy: primary triangulation or trav¬ 
erse; secondary triangulation or traverse; and tertiary trav¬ 
erse and meander for the location of details. 

Third. The measurement of heights, which usually ac¬ 
companies the horizontal location and may be similarly di- 


ORGANIZATION OF FIELD SDR FEY. 


23 


vided into three classes, dependent upon their degree of 
accuracy. 

Fourth. The sketching of the map. 

If the area under examination is small or the scale be of 
topographic magnitude, the first of the foregoing operations 
may be omitted, when the topographic party will have (1) 
To determine the horizontal positions of points; (2) To 
measure the heights of these points; and (3) To sketch in the 
map details as controlled by the horizontal and vertical loca¬ 
tions so procured. 

Where map-making is executed for geographic or explora¬ 
tory purposes and on a small scale in open triangulation coun¬ 
try, as that in the arid regions of the West, the skilled force 
may consist of only the topographer in charge. Where the 
map scale is increased up to topographic dimensions or the 
country is hidden from view by timber or because of its lack 
of relief, the topographer may be assisted by one or more 
aides whose functions will be variously performed according 
to the conditions of the country. 

12. Surveying Open Country. —In making a geographic 
map on scales varying, say, from one-half mile to four miles to 
the inch in open, rolling, or mountainous country suited to tri¬ 
angulation, all sketching and the execution of the plane-table 
triangulation (Chap. IX) or other control should be done by 
the topographer in charge. He may be aided by one to three 
assistants according to the speed with which he is able to 
work and the difficulties encountered by the assistants in 
leveling (Chap. XV). It is assumed that the topographer has 
a fixed area to map, and that within this area he is in posses¬ 
sion of the geodetic positions (Chap. XXIX) of two or more 
prominent points and the altitude of at least one. 

With the positions of these points platted on his plane- 
table sheet (Art. 188) he proceeds, as outlined in Article 54, 
to make a reconnaissance of the area for the erection of sig¬ 
nals and to locate prominent points on summits and in the 




SURVEYING FOE SMALL-SCALE MAES. 


lower or drainage lines of the country by plane-table triangu¬ 
lation (Art. 73). Meantime, one assistant may be running 
lines of spirit-levels (Chap. XV) for the control of the verti¬ 
cal element, while one or two assistants are making odometer 
(Art. 98) or stadia traverses (Chap. XII) of roads or trails for 
the control of the sketching and the mapping in plan of the 
roads and streams. This preliminary control executed, the 
topographer adjusts toJiis triangulation locations the traverses 
run by the assistants (Art. 81), and writes upon them in 
their proper places the elevations determined by leveling or 
or vertical angulation (Chap. XVII). 

In Fig. 1 is shown a typical triangulation control sheet , 
the directions of the sight lines being indicated so as to show 
the mode of derivation of the various locations. The stations 
and located intersection points are numbered in order to show 
the sequence in which they were procured. The traversing 
executed for the same region is illustrated in Fig. 2, from 
which it will be seen that merely the plans of the roads with 
their various bends, stream crossings, and the houses along 
them were mapped. Hill summits and other prominent ob¬ 
jects to one side or other of the traversed route were inter¬ 
sected (Art. 84) in order to give additional locations and to 
facilitate the adjustment of the traverse to the triangulation. 
The closure errors of the various traversed circuits are shown, 
and an inspection of these makes it clear that in every case 
the errors in traverse work are so small as not to affect the 
quality of the control, because the adjustment of the traverses 
by means of points on them which are located by the plane- 
table triangulation will distribute the errors in the various 
road tangents in such manner as to make them imperceptibly 
small on the resulting map. The product of such adjustment 
is shown on Fig. 3, which is the base on which the topog¬ 
rapher begins his sketching. On this sketch sheet are the 
locations obtained by him in the execution of his plane-table 
triangulation, the traverses as adjusted to this control, and 


SURVEYING OPEN COUNTRY . 


25 



Scale flg-g-ffX)* 














26 


SURVEYING FOR SMALL-SCALE MARS . 



Frostburg, Md. 
Scale 625W 















TOPOGRAPHIC CONTROL. 


27 



Fig. 3 . —Adjusted Sketch Sheet. Frostburg, Md. 

Scale 











28 


SURVEYING FOE SMALL-SCALE MAPS. 


elevations from vertical angulation or spirit-leveling written 
in their appropriate places. 

If the work be the making of a topographic map on scales 
larger than those above described, and the country be still of 
the same topographic character—namely, open, with salient 
summits,—a system of control similar to the above must in like 
manner first be executed by the development of plane-table 
triangulation and the running of control, traverse, and level 
lines. But the after-work of sketching the map will be con¬ 
ducted in a different manner than for the smaller scales, 
because of the greater detail required, the shorter distances 
to be traveled by the topographer in performing the work, 
and his consequent nearness to the various features which he 
is to map. 

13. Sketching Open Country. — Having the control 
platted on the sketch sheet as shown in Fig. 3, and where 
roads are sufficiently abundant to cut up the map with trav¬ 
erses so near one to the other that the topographer may not 
have to sketch more than one-half to one inch to either side 
of his position, the sketching of the topography proceeds as 
follows: 

Taking the sketch sheet on a board in his lap, the topog¬ 
rapher for cheapness and convenience, because of the speedy 
drives over every road. Where these are not sufficiently near 
one to the other he walks in between them, pacing distance 
(Art. 95), and getting direction by sighting fixed objects, 
while he sketches the plan of the contour lines (Art. 193) as. 
far as he can safely see them to either side of his path. This 
operation is performed by setting out from such fixed points 
as a road junction, a located house, or a stream crossing, the 
position of which is platted on his map and the elevation of 
which is known. Adjusting the index of his aneroid at the 
known elevation (Art. 176), he drives along, keeping the 
platted direction of the road parallel to its position on the 
ground and marking on the map the positions at which the 


SKETCHING OPEN COUNTRY. 


2 9 


various contours are crossed by his route. Thus, if his contour 
interval be twenty feet, at every change of twenty feet as indi¬ 
cated by the barometer he stops, and, knowing his position 
on the map either by reference to bends in the roads, houses, 
or by having counted the revolutions of his wheel from a 
known point, he glances along the trend of the slopes to one 
side or the other, following by eye the level line of his con¬ 
tour, and this he sketches in horizontal plan upon the map. 
At first he may be aided in this by a hand-level (Art. 156), 
but as he acquires skill with practice he is able to estimate 
the position and direction of the contour line to either side 
with great accuracy, and finally to interpolate other contours 
above and below that on which he is placed with such preci¬ 
sion as not to affect the quality of the resulting map by a con¬ 
tour interval. 

The aneroid being an unreliable instrument, he must not 
drive more than two or three miles without checking it at a 
well-determined elevation. This he is usually able to do at 
houses, or hill-summits, or other points the positions of which 
have been determined by his prior control. If he is not able 
so to check his aneroid, he hastily sets up his plane-table, 
reads with the telescopic alidade a few vertical angles (Art. 
162) to hilltops or houses in sight and the elevations of which 
are known, and, with these angles and the distances which he 
can measure from his position to the points sighted as shown 
on the adjusted control, he is at once able to compute the 
elevation of his position (Art. 161) within two or three feet 
and thus check his aneroid. At the same time he is in simi¬ 
lar manner able frequently to throw out other elevations by 
sighting from the position thus determined to houses or sum¬ 
mits near by which may have been located by the traverse 
(Art. 84), and the heights of which he determines now from 
his angulation. The topographer thus sketches the whole 
area assigned him, not only mapping the contours, drainage, 
political boundaries, and other topographic features, but also 


30 


SURVEYING FOR SMALL-SCALE MAES. 


checking the positions of houses and summits and the direc¬ 
tions and bends of roads and streams as located by the trav- 
erseman (Fig. 4). 

Where the hills are more prominent and the slopes bolder 
and steeper, the topographer sketches these from his various 
view points by mterpolating contours between the located con¬ 
trol points. With the sketch-board in his lap or on the tripod 
and approximately oriented, looking about in various direc¬ 
tions at hill-summits, houses on slopes, spurs, etc., which 
may with their elevations be platted on his map, he first 
sketches in plan the streams and drainage lines as well as the 
directions of slopes. Then he sketches the position of con¬ 
tour lines about such control points as summits, salients, and 
his own position. With these as guides he is then unable to 
go astray in the interpolation of the intermediate contours 
which complete the map of the area immediately about him. 

The sketching of the topography for large-scale maps dif¬ 
fers rather in degree than in kind from the above. The large- 
scale map covering as it does a relatively small area, the to¬ 
pographer is not under the necessity of traveling with such 
speed as to necessitate his using wheeled conveyance. At 
the same time the largeness of the scale places the roads at 
much greater distances apart on the map and necessitates his 
traveling between these to greater extent. It will thus 
be seen that the scale and the ability to travel over the coun¬ 
try work harmoniously one with the other. For the smaller 
geographic scales the roads are so close together on the map 
as to afford sufficient control and sufficient number of viewing 
points for sketching the topography of the average open 
country, whereas on large-scale topographic maps these 
roads are in plan much farther apart, but the time consumed 
in walking between them is a comparatively small item be¬ 
cause of the decrease in the distances to be covered. 

In sketching a large-scale map the topographer will have 
about the same relative amount of primary control as above 








i 


Fig. 4.—Completed Topographic Map, Frostburg, Md. 

Scale 1 to 62,500. Contour interval 20 ft. 











































\ 



m 


SURVEYING WOODLAND. 


33 


indicated. Starting out with some known point and on foot, 
accompanied by one or more stadiamen, he sets up and ori¬ 
ents his plane-table, and, having considerable areas to fill in 
on his map between his present position and his next recog¬ 
nizable natural feature, he posts the stadiamen at convenient 
changes in the slope of the country or at houses or trees or 
bends in the streams, and drawing direction lines and reading 
distances by stadia to these positions he obtains additional 
locations to control the sketching (Art. ioi), which is exe¬ 
cuted as above described. In the progress of this work he 
not only determines horizontal positions by sighting to the 
rods held by his stadiamen, but also the vertical positions 
of the same points (Art. 102). For very large-scale maps 
and under some conditions the work may be expedited by 
permitting the assistants to sketch the contours immediately 
adjacent to their stadia stations, and these sketch notes must 
be given the topographer at frequent intervals to be trans¬ 
ferred to his map. In this manner one topographer may 
handle from one to three stadiamen, providing he uses judg¬ 
ment in the selection of his and their positions. Forsmaller- 
scale topographic mapping the work may be expedited by the 
stadiamen riding on horseback from one position to another, 
or even by the topographer himself using this means to get 
about. 

14. Surveying Woodland or Plains. —The securing of 
control in densely wooded country, as that of the Adirondack 
region or the woods of Minnesota, Michigan, and of Washing¬ 
ton ; or the securing of control for very flat plains country, 
as that of the Dakotas and Nebraska, is accomplished by dif¬ 
ferent means than must be adopted in open triangulation 
country. Be the scale of the resulting map large or small, the 
primary control may be obtained most economically either 
by triangulation or by traverse methods. If the country is 
wooded and rolling , it may be more economical to clear the 
higher summits or to erect high viewing scaffolds upon them, 



34 


SURVEYING FOR SMALL-SCALE MARS . 


from which to conduct a skeleton plane-table triangulation. 
Intermediate positions may be obtained by placing signal- 
flags in tall trees and locating these by intersection or using 
them to obtain other positions by resection. With prac¬ 
tice the topographer will thus triangulate the most forbid¬ 
ding woods country more expeditiously than it could 
otherwise be controlled, by taking advantage of every out¬ 
look, as a rock on a hillside, a lake, a small clearing for a 
farm, or by clearing or signaling the commanding summits. 
H e will thus occupy only such points as those just described, 
locating by intersection (Art. 73) from them the flags on the 
more wooded and forbidding ones which may be the more 
commanding positions, and using the latter again for carrying 
on his work by resection (Art. 74). 

In level plains or in wooded plateau land the control 
may of necessity be executed only by traverse methods. In 
such case where the scale is of geographic dimensions one or two 
astronomic stations should be determined (Part VI), or for 
larger scales it may suffice to assume the initial position. From 
this primary traverse lines should be run (Art. 22-6) at con¬ 
siderable distance one from the other, depending upon the 
scale. For the one-mile scale a nearness of fifteen to twenty 
miles will suffice. For the two-mile scale these primary traverse 
lines may be double the distance apart; for a large topographic 
scale a relatively smaller distance, depending upon the map 
scale; for all scales a distance corresponding to fifteen to 
twenty-five inches on the map according to the topography. 

Between these primary traverse lines others of less accu¬ 
racy should be run as a secondary control. On these dis¬ 
tances should be measured by wheel (Art. 98) when the 
vehicle can be driven in straight tangents, by stadia (Art. 
101) in open irregular country, or by chain (Art. 99) or tape 
(Art. 97) through underbrush or dense wood. Elevations 
will be secured in the woods by vertical angulation to stadia 
(Art. 102) or by spirit-leveling (Chap. XV); in the open 


35 


SKETCHING PLAINS LAND 

or plains by vertical angulation to fixed objects, as the 
eaves or chimneys or window-sills of houses, the platforms of 
windmills, etc. (Art. 160), or to the stadia-rod, as well as by 
spirit-leveling. The secondary traverse is usually executed 
by the party chief while his assistants are engaged in tertiary 
traverse for the filling in of topographic details or the procur¬ 
ing of vertical control. 

The primary and secondary control having been procured 
as above, this should be platted on sketch sheets of the cus¬ 
tomary large plane-table size for open country (Art. 68), and 
preferably in small detached pieces placed on small boards of 
about six inches square, where the latter have to be carried 
through woods and underbrush. These control sheets will be 
not dissimilar to those described in Article 13, excepting that 
they will lack the location of points procured by angula¬ 
tion and will consist almost wholly of platted traverse lines. 
In order that the topographer when sketching may identify 
these lines on the ground, trees must be frequently blazed 
in woods when the traverses are being run and station num¬ 
bers or elevations be written on the blazings. 

15. Sketching Woodland or Plains.—With the control 
platted on the sketch sheet as just described, the topographer 
in plains work starts out and drives over the country much as 
described in Article 13, traveling over all the traversed roads and 
checking his aneroid by setting in at known elevations or by 
angulation to and from buildings and similar objects. As the 
country is relatively flat, the contour lines are at considerable 
distances apart in plan, and consequently a very small differ¬ 
ence in vertical elevation makes a considerable change in the 
horizontal location of a contour. Therefore the determina¬ 
tion of the vertical element should be of greater relative accu¬ 
racy, that the resulting map may be correct. 

In the woods the sketching is executed in an entirely dif¬ 
ferent manner. Little skill is required in the depiction of 
the topography, as it is impossible to see the country and 


SURVEYING FOR SMALL-SCALE MAES. 

therefore to sketch it in the ordinary sense. The topographer 
is limited to sketching that which is directly under foot—in 
other words, to mere contour crossings—and in order that 
these may be connected the traverses must be much nearer 
together, and not only the topographer but his more skillful 
assistants are all engaged in sketching and traversing at the 
same time. Starting out with the primary and secondary 
control as obtained in the last article, the topographer travels 
over those traversed routes which have been blazed and 
sketches the contours upon these while his assistants run addi¬ 
tional traverses over controlling routes, as along stream beds 
and ridge crests, and so close together as to completely 
command all the country under foot. These traverses will 
be of crude quality, directions being obtained by sight alidade 
(Art. 62) and traverse-table (Art. 61), and distances by pacing 
(Art. 95) or by dragging a light linen tape (Art. 97). Each 
day the topographer must adjust to his control sheet the 
traverses with accompanying sketching as executed by his 
assistants. With such a skeleton of topography on highest 
and lowest lines, i.e., contour crossings of streams and ridges, 
he can readily interpolate contours for most of the inter¬ 
mediate spaces and, following after his assistants, fill in those 
places which are not fully mapped. 

In the execution of a survey under such conditions the 
topographer’s work is largely supervisory and consists chiefly 
in the management of the work of his assistants, the adjust¬ 
ment of their sketching, and its inspection as he fills in the 
details omitted by them. There is little room for them to go 
astray, because they only sketch that which they walk over. 
The topographer should invariably reserve for himself the 
higher ridges, the ponds, and the more open places in order 
that quality and speed may be obtained by the utilization of 
his skill in that work which gives some opportunity for sketch¬ 
ing at a distance from the traveled route. 

16. Control from Public Land Lines_In the western 



CONTROL FROM PUBLIC LAND LINES. 


37 


United States where the public land surveys have been exe¬ 
cuted in recent years and with sufficient accuracy to furnish 
horizontal control, this may come almost wholly from the 
township and section plats filed in the United States Land 
Office. The topographer takes into the field paper on which 
sections and quarter sections are ruled and numbered. On 
these he writes at the proper section corners the elevations 
as determined from the primary spirit-levels (Chap. XV). He 
also indicates on the northern and western margins of each 
township the offsets and fractional sections as shown on the 
published land plats (Fig. 5). At some period during the 
progress of field-work the topographer adjusts the land-line 
work to positions determined either by primary triangulation 
(Chap. XXV) or traverse (Chap. XXIII), supplementing this 
by additional control where necessary. 

17. Sketching over Public Land Lines.—With the con¬ 
trol sheet prepared as described in the last article, the topog¬ 
rapher proceeds to drive over the section lines on which roads 
have been opened. The control sheet is attached to a plane- 
table board. Starting from a known section corner, he drives 
in a straight line down one of the section lines to other sec¬ 
tion corners, determining his position by counting revolutions 
of the wheel (Art. 98) and sketching contour crossings as he 
progresses. 

Starting out with a known elevation from spirit-levels 
(Chap. XV), he determines other elevations as he proceeds by 
setting up his plane-table at a section corner or opposite a 
house which he can locate by odometer distance, and reads 
vertical angles from the point of known elevation to houses, 
windmills, or other objects in sight (Art. 162), drawing direc¬ 
tion lines to them as an aid in their identification (Art. 84). 
Driving on until he comes to one of these objects and being 
thus able to locate it on his plane-table, he measures the dis¬ 
tance from it to the point from which the angle was taken and 
at once computes his elevation (Art. 161). Or, setting up his 


3$ SURVEYING FOR SMALL-SCALE MAPS . 



Fig. 5 . —Land Survey Control for Topografhic Sketching. 


North Dakota. 

Original scale 2 inches to 1 mile. 























SKETCHING OVER PUBLIC LAND LINES . 


39 


plane-table board from some known position, as determined 
from his section lines and odometer, he reads vertical angles to 
houses or windmills, the heights of which have already been 
determined by vertical angulation, and thus brings down to 
his present position an elevation by means of the angle read 
and distance measured on his board. In conducting vertical 
angulation in this manner care must be taken to sight at some 
well-defined point, as a platform or top of a windmill, the 
gable or top of a house or top of door-sill, etc. 

As the sketching is a comparatively simple process under 
these conditions because of the flatness of the terrane, his 
work may be expedited by permitting his more skillful assist¬ 
ants to aid in sketching. In order that he may control their 
work he drives and sketches over those roads which parallel 
the roads of his assistants on either side, and in such manner 
obtains a clear insight into the work performed by them. 
The assistants may determine elevations either by vertical 
angulation, as does the party chief, or by aneroid frequently 
checked, say at distances not exceeding two miles between 
the better elevations obtained by the topographer. On such 
a sketch sheet as it comes from the plane-table board (Fig. 6) 
the roads have been clearly marked over the section lines and 
additional diagonal roads have been traversed or sketched 
directly on the plane-table board, controlled by section cor¬ 
ners, the outlines of lakes having been obtained by stadia 
(Art. ioi). 

Where the topographic map is made at the same time as 
the subdivision of the public lands, as was the case in the 
Indian Territory surveys of the United States Geological 
Survey, the cost of executing the topographic survey scarcely 
exceeds the cost necessarily involved in making the land sub¬ 
division or cadastral survey. The only additional cost in the 
execution of the topographic survey is that for leveling. 
Fig- 33 is an example of the cadastral map resulting from 
such a survey of the public lands. The topographic map of 


40 


SURVEYING FOR SMALL-SCALE MARS . 


the same region corresponds in appearance almost identically 
with that shown in Fig. 6, being shorn of the various sym¬ 
bols used on the Land Survey Maps. 

18. Cost of Topographic Surveys —As indicated in 
Tabl es i. n , and III, the cost of topographic surveying 
varies widely with the character of the country, the scale of 
the map, and the contour interval. Such topographic surveys 
as are executed by the United States Geological Survey range 
in cost for maps of a scale of one mile to the inch and 20-foot 
contour interval, similar to those described for open country 
in Articles 12 and 13, from $10.00 to $20.00 per square mile. 
Such as are described in Articles 14 and 15, for plants or 
woodland , range in price from $8.00 to $12.00 per square mile 
for the former to between $15.00 and $30.00 for the latter. 
The highest-priced work of this kind which can be executed 
being the woodland survey, and the cheapest country to map 
topographically being the open plain. 

Land-survey country , as that instanced in Article 16, which 
is a survey of a portion of North Dakota, ranges in cost from 
$5.00 to $8.00 per square mile, where the topographic map is 
made on a scale of two miles to one inch and in 20-foot con¬ 
tours. For the same scale and in mountainous country, as 
that of the South and West, the cost is from $8.00 to $12.00 
per square mile. 

If any endeavor is made to do work for othe** purposes 
than the procurement of a topographic map, as for che deter¬ 
mination of land lines or the staking out of canals or railroads, 
the cost of the survey is at once greatly enhanced. It is this 
which has added so greatly to the relative cost as shown in 
the tables cited of some private topographic surveys as well 
as of the cadastral surveys. 

19. Art of Topographic Sketching —Mr. A. M. Well¬ 
ington aptly said of topographic surveying that “ the sketch¬ 
ing of the form of the terrane by eye is truly an art as 
distinguished from a science, which latter, however difficult it 


Fig. 6.—Topographic Map on Land Survey Control, near Fargo 
Scale i to 125,000. Contour interval 20 ft. 












































































































































* 



















THE ART OF TOPOGRAPHIC SKETCHING . 


43 


may be, is always susceptible to rigorous and exact analysis. 
An art, on the other hand, is something which cannot be 
taught by definite, fixed rules which must be exactly fol¬ 
lowed, though instruction may be given in its general prin¬ 
ciples.” 

In representing the heights and slopes of a given piece of 
country by contour lines, every case presents some peculiari¬ 
ties which must be met, as they are presented, by the topog¬ 
rapher’s own resources. No hard-and fast limit of minute¬ 
ness of detail can be previously fixed. The scale chosen for 
the topographic map limits this to a certain extent, but its 
exact limits must be set by the topographer’s own experience 
and good judgment, that he may discriminate between impor¬ 
tant and trifling features; those which are usual and common 
to the region being mapped, and those which are accidental 
or uncommon, and which should therefore be accentuated. 
Above all, the topographer must exhibit an alertness to dis¬ 
tinguish as to what amount of detail should be omitted and 
that which should be included. Hesitancy in this is the chief 
source of slow progress. Valuable time may be wasted in 
the representation of features which may be lost on the scale 
of the work and which are common in all localities to the 
topographic forms being sketched ; while features characteristic 
of such special topographic forms as those produced by erup¬ 
tion, erosion, or abrasion, or those indicative of the structure 
of the region and which give distinctive character to its topog¬ 
raphy, may be lost sight of or be covered up in the map by 
too careful attention to minute details. 

The characteristic features of a terrane are best observed 
from a point nearly on the same level; and as between 
sketching features from above or below for a reasonable 
range, sketching from below is the better, as features viewed 
from any considerable height above are apt to appear dwarfed 
and much detail of undulation of the surface lost sight of. 
Yet, as a precise representation of the land requires sketch- 


44 


SURVEYING FOR SMALL-SCALE MAES. 


ing its forms from numerous positions at intervals not far 
apart, the necessity will rarely arise of observing surface 
forms from points of observation much above or below the 
surface represented, excepting in case of very small scale 
geographic or exploratory surveys. 

20 . Optical Illusions in Sketching Topography. —In 
sketching topographic forms by eye there are a number of 
optical illusions to which it is well to call attention, though 
the effect of these can be entirely overlooked in the sketching 
of detailed topography such as would be mapped on scales 
less than one mile to the inch. But for the sketching of 
topographic maps on smaller scales, where the eye has to be 
more depended upon, these illusions become more important. 
Most of these have been well classified by Mr. A. M. Welling¬ 
ton in his admirable work on railway location, and they are 
here summarized, with variations, from that work. Among 
the more serious of such illusions are the following: 

1. The eye foreshortens the distance in an air line and 
materially exaggerates the comparative length of a lateral off¬ 
set so as to greatly exaggerate the loss of distance from any 
deflection. 

2. The eye exaggerates the sharpness of projecting points 
and spurs, and accordingly exaggerates the angles. 

3. In looking, however, at smooth or gentle slopes from 
a distance, the tendency of the eye is to decrease the angle so 
that in such country as the rolling plains of the West slopes 
look much gentler , the inclinations much less, than they are in 
fact. 

4. In this connection the eye is liable to make slopes 
looked at from a distance appear steeper and higher than they 
are in fact, when they are compared with known slopes and 
elevations of lesser dimensions near by. 

5. Again, the unaccustomed eye, which mentally meas¬ 
ures all dimensions by referring them to those with which it 
is acquainted, is apt to make a divide or pass appear lower 


OPTICAL ILLUSIONS IN SKETCHING TOPOGRAPHY. 45 

than a nearer divide or pass to which it is referred in one 
sweep of the vision, whereas it may be higher (Fig. 7). 



Fig. 7 . —Optical Illusion as to Relative Heights of Divides. 

A is nearer and lower than B. 

6. The eye invariably exaggerates the steepness of the slopes 
of mountains, these appearing to have inclinations of from 60 
degrees to almost vertical, whereas in fact the steepest slopes 
are rarely as great as 45 degrees. 

7. The eye trained to estimate slopes and distances in 
regions of large topographic features—that is, regions of ex¬ 
treme relief or differences of elevation—will be at a disadvan¬ 
tage in making similar estimates in a country in which the 
differences of elevation are small. The tendency of one 
accustomed to estimating the topographic forms in the Rocky 
Mountains, where differences of elevation and distances visible 
to one sweep of the eye are great, will be to overestimate 
heights and distances in the less rugged country of the Eastern 
States, where great detail in topography exists, and thus de¬ 
ceives the eye into an exaggerated notion of the amount of the 
relief. 

8. In viewing the terrane with an idea of estimating its 
roughness as affording a possible route for railways, canals, or 
similar works, a rugged mountain gorge with occasional pre¬ 
cipitous narrows, separated by river flats, may appear much 
more difficult and much rougher than it is in fact. This is 
especially so as compared with a gently undulating or rolling 












46 


SURVEYING FOR SMALL-SCALE MARS. 


country, which, when viewed from a distance, appears to be 
comparatively level, while a nearer view will show it to be full 
of elevations or depressions which will render construction 
most expensive, because of the rapid and numerous succession 
of large cuts and fills. 

The effect on the eye and the mind is to exaggerate the 
ruggedness of a country which is difficult to travel because of 
such impediments as broken stone, fallen timber, creeks, and 
swamps, whereas a region where travel is easy and free, as in 
open rolling plains country or where good roads abound, is 
often estimated to be much simpler and more level topo° 
graphically than is the other region. 


CHAPTER III. 


SURVEYING FOR DETAILED OR SPECIAL MAPS. 

2 io Topography for Railway Location. —Some of the 
worst errors in engineering location originate in reconnais¬ 
sance, for the reason that the average reconnaissance sur¬ 
veys are not of areas, but of routes or lines, and there is great 
danger of serious error in the selection of the line to be sur¬ 
veyed. It may, accordingly, be stated that a railway recon- 
naissance should not be of a line, but of an area sufficiently 
wide on each side of an air line between the fixed termini to 
include the most circuitous routes connecting these. The 
results of such a survey should be embodied in a topographic 
map of greater or less detail, according to the nature and ex¬ 
tent of the country. If the reconnaissance be of a great rail¬ 
road, such as some of the Pacific roads, built through hun¬ 
dreds of miles of unknown country the resulting map should 
be on a small scale, perhaps 2 to 4 miles to the inch, and 
with contour intervals varying from 20 to 100 or 200 feet, 
according to the differences of elevation encountered and the 
probable positions of several locations. With such maps as 
those of the U. S. Geological Survey, the number of possible 
routes may be reduced to two or three, and a more detailed 
topographic survey should then be made of these on which to 
plan the final location. 

As ordinarily practiced, topographic surveys for railways 
are made by the older methods, with transit and chain 01- 
stadia and with spirit-level; notes of the surveys are kept with 
accompanying sketches in note-books, and these are reduced 
to map form in the office. The same results can be much 

47 


48 SURVEYING FOR DETAILED OR SPECIAL MAPS. 


more satisfactorily and more rapidly procured by using the 
plane-table in place of the transit, and the resulting map, 
bdng plotted in the field, is a more accurate and avai’able 
representation of the terrane than can possibly be made from 
plotting notes in an office. 

The Germans , who are very thorough in taking topography 
for railroads , divide the work into three separate surveys of 
different degrees of accuracy: first, recourse is had to the 
government topographic maps on a scale of approximately 
i: 100,000, and on this a preliminary route or routes are laid 
down: second, a more detailed topographic survey is made in 
the field on a scale of I : 2500 as a maximum or 1 : 10,000 as 
a minimum, with contour lines of 15 feet interval. This map 
is limited in area from a few yards to a few hundred yards in 
width, according to the nature of the country. Where no 
previous small-scale topographic survey exists, the base 
of this more detailed or second survey is a transit (Art. 87) 
or plane-table (Art. 83) and level (Art. 129) traverse, follow¬ 
ing as nearly as possible the approximate route of the pro¬ 
posed railway. Bench-marks (Art. 135) are established along 
this at distances of from 500 to 1000 feet, by which the ane¬ 
roid may be checked. With this transit line completed on 
the proper s< ale, the topographer goes over the ground and, 
by means of distances from pacing (Art. 95) or odometer 
(Art. 98), and elevations by aneroid (Art. 176), constructs a 
hasty contour map on which are indicated all roads, water¬ 
courses, structures, high-water marks of bridges, width and 
height of existing bridges and culverts ; and all other necessary 
topographic details as to the position of rock masses, strike 
and dip of strata, swamps, springs, quarries, etc. 

On such a map as this, hastily and cheaply made, it is 
possible to plan the detailed topographic map, limited from a 
few yards to 100 or 200 yards in width and covering what 
will practically be the final route of the located line as 
obtained from the second survey. This final detailed survey, 


DETAILED TOPOGRAPHIC SURVEYS. 


49 


from which the paper location is to be taken, should be on 
a scale of from I : 500 up to 1 : 1000 and with contours of 
about 5 feet interval, more or less, according to the nature 
of the land. There is plotted on the plane-table sheet the 
transit and level base line previously run for the second 
survey, and the instruments now used by the topographer are 
of a more accurate nature, consisting of a plane-table (Arts. 
58 and 83) for direction and mapping, two or more stadia 
rodmen for distances (Art. 102), while elevations are had 
by vertical angles with the alidade (Art. 59). On this final 
map are shown much the same topographic details as on the 
second, but all are more accurately located and the eleva¬ 
tions are of a more refined nature. The data furnished by 
this final map will serve all the purposes of making a last 
paper location of the line, from which the engineer will in 
the field possibly deviate according to the appearance of the 
route traveled as presented to his eye when the location is 
laid down. 

Mr. Wellington’s location of the Jalapa branch of the 
Mexican Central Railway (Fig. 8) is an excellent example of 
a detailed contour topographic map for railway location. 
This was platted in the field on the scale of 1 : 1000, or about 
83-J feet to 1 inch. The contour interval was 2 meters, or 
6.56 feet. 

22. Detailed Topographic Surveys for Railway Loca¬ 
tion. —Prior to making the location, which may be made in 
part from the notes of preliminary surveys, a narrow belt of 
topography should be mapped in detail, its width being re¬ 
stricted as far as possible, providing the preliminaries have 
been skillfully conducted or have been preceded by a small- 
scale topographic map executed with especial care along the 
possible routes of the location (Art. 21). On the detailed 
topographic map a paper location may be made, from which 
full notes of the alignment can be derived, the points of curve 
and tangent taken off, and a profile of the paper location pre- 


50 SURVEYING FOR DETAILED OR SPECIAL MAPS. 



Pig, 8.—Contour Topographic Survey for Location of Mexican Central Railway. 
Scale of original 83^ ft. to I inch. Contour interval 6,5 ft. 











































































DE TAILED TOPOGRAPHIC SUR VE VS. 


51 


pared, For the making of the paper location the topography 
should be as exact and the contour lines should be as accu¬ 
rately placed as the scale of the map will permit, in order that 
a line may be located upon the map and a profile called off 
from it which shall agree as closely as possible with the sub¬ 
sequent transit locati m and spirit-level profile. In running 
the field location from a paper location, the projected profile 
and not the projected alignment must be run. 

In making such a map it is neither necessary nor possible 
to locate every point on each contour, the horizontal and ver¬ 
tical locations of the contours being at such distances apart 
that their projections on the map will be so close together 
that in connecting them by eye in the field the topographer 
cannot go astray by an appreciable distance. With a detailed 
contour map made as described for the location of canals (Art. 
23), a grade contour or location line may be drawn which will 
show where the plane of the roadbed will cut the natural sur¬ 
face and from which it will at once be seen whether or not the 
location is the most favorable the topography will permit. 

The error into which many have fallen is in assuming too 
much or too little for the topography as a guide to location. 
The topographic map fails to show many essentials requisite 
in making a location, as it gives no evidence of the materials 
to be encountered, nor does it convey an adequate idea of the 
magnitude of the excavations and fills. The topographic 
map must be supplemented by a careful visual reconnaissance 
of the line which it covers. Such topography should there¬ 
fore be restricted in its width and amount, and no attempt 
should be made to make a final location from such a map. On 
the other hand, where a topographic map is not made, and too 
much reliance is placed on the visual reconnaissance of the 
country, the greatest errors are at once introduced in encoun¬ 
tering a bad system of gradients, in overlooking important 
towns, or in otherwise selecting inappropriate routes. 


52 SURVEYING FOR DETAILED OR SPECIAL MAPS. 

In planning the location on a detailed topographic map, 
the engineer should begin at a summit or similar fixed point, 
assuming or taking from a guide-map an initial elevation. 
Then with a pair of dividers he should step off such distances 
that these will correspond to the grade chosen and their 
termini end on the map above or below such contours as will 
give the proper differences in elevation to produce such grades. 
By this means a grade contour can be sketched in on the map 
and then connected by tangent lines. The latter must, in 
turn, be connected by throwing in curves the radii of which 
shall be as large as possible, care being taken that the grades 
on these shall be properly compensated. With such a paper 
location it is then possible, by means of scale and protractor, 
to take off the directions and distances in a note-book, when, 
with these as a guide, the located line may be run on the 
ground and changed or modified in the field as the visual 
observation of the engineer may suggest. 

Speed in mapping railway topography varies greatly with 
the scale selected and the character of the land mapped. 
One party working in flat, desert country in Utah ran 20 
linear miles in a day of 9 hours, including running of spirit- 
levels. The same party working later in mountainous country 
in Washington averaged during a long period of time less than 
\ mile a day, in one instance working six weeks on a location 
through i-J miles of canyon. A party working on railway 
location and mapping topography on the plains of Kansas 
made an average speed of 2.1 miles a day at an average cost, 
including all expenses, of $11.03 P er linear mile. The Utah 
work averaged about $2.50 per mile, and the cost of much of 
the Washington work exceeded $100.00 per linear mile. 

23. Topographic Survey for Canal Location. —Surveys 
for canal lines or lines of conduits, etc., are best made by having 
the leveling (Chap. XV) precede the plane-table or transit 
work. The level will then run out a grade contour having the 
requisite fall per mile, and the transit (Art. 87) or plane-table 


TOPOGRAPHIC SURVEY FOR CANAL LOCATION. 53 

(Art. 83) with chain measurements (Art. 99) will follow the « 
level, locating this grade contour. Topography may be 
taken on either side by stadia (Art. 101) and plane-table so 
that in the final location of the canal the preliminary grade 
contour may be shifted to suit the sketched topography, 
much as the line of a railway location would be shifted from 
similar data (Art. 22). 

An interesting example of a detailed topographic survey 
for the final location of an irrigation canal is that made by Mr* 

J. B. Lippincott of the Santa Ana Canal, through a rocky can¬ 
yon. This location was made upon a carefully prepared 
topographic map drawn on a scale of 50 feet to 1 inch, with 
contour interval of 5 feet. The maps were plotted from cross- 
section notes based on two connected and approximately par¬ 
allel preliminary lines, the contour curves being sketched in the 
field to indicate intervening irregularities of surface. The 
preliminary controlling lines were carefully run with transit 
and chain, were frequently connected, and had a vertical in¬ 
terval of 70 feet. The space between and for thirty feet 
above the upper line, or for a total of 100 feet vertically, was 
carefully contoured. From the map thus prepared a more 
accurate cross-sectioning was made, and from these notes a 
new contour map of the ground was prepared on a scale of 30 
feet to an inch over the more difficult portions of the line, 
after a preliminary location had been selected on the first con¬ 
tour map. Fig. 9 gives a plat of one of the roughest por¬ 
tions of this line, and on it are shown in small circles the 
various points located on each contour. The plane-table was 
used and was set up generally as shown by the station num¬ 
bers and triangles on the preliminary and plotted traverses, 
and directions were measured to stadia-rods held at various 
points on the io-foot contour lines (Art. 101). The posi¬ 
tions of the contour lines at these points were therefore 
plotted, and the corresponding elevations were immediately 
connected as contour lines on the plane-table sheet. In this 


54 


SURVEYING FOR DEI'AILED OR SPECIAL MAPS * 



* 


Fig. 9.—Detailed Contour Survey for Canal Location. 
Original scale 100 ft. to 1 inch. Contour interval 2 ft. 































TOPOGRAPHIC SURVEY FOR CANAL LOCATION. 55 

way enough points were located on each contour to sufficiently 
control it, and the immediate 2-foot contour lines were in¬ 
terpolated by eye estimation in the field. 

In doing this work three various methods were tried: (i), 
by locating the contour lines with slope-board and rod; (2), 
by locating the contours at right angles to the stations occu¬ 
pied by a levelman using a hand-level (Art. 156); and (3), 
by means of the plane-table and stadia (Art. 101). Mr. Lip- 
pincott says that as a result of these tests there is no question 
between the quality of the three classes of work; that without 
plane-table the work had to be plotted up in office and located 
points connected by estimation or from rough sketches; with 
the plane-table the same points were plotted immediately, 
in the field, and the connections between these made with the 
terrane in view, and that the resulting map by plane-table 
much more accurately expressed the slopes of the land than 
did the maps made by the other methods. The speed by the 
various methods was about the same. The party consisted 
generally of five persons, including the topographer, levelman, 
and rodman, and the speed was from 2500 to 4000 linear feet 
per day, actually locating four io-foot contours and sketching 
in five or six more, a total of 100 feet vertical interval, and 
interpolating the 2-foot contours. Where side canyons and 
ravines were passed the slope-board was found to be entirely 
inadequate and helpless, while by the use of levelman and 
hand-level without the plane-table, and with taped traverse 
lines, the conditions were improved, but the work was of 
the crudest character so far as its topographic expression was 
concerned. 

An example of a preliminary topographic survey of a 
canal line , made under the author with plane-table and on a 
small scale to determine the possibility of bringing the water 
from a stream or reservoir to certain lands for purposes of 
irrigation, is illustrated in Fig. 10. The scale of this illus¬ 
tration is denoted by the land section lines, each section 


56 SURVEYING FOR DETAILED OR SPECIAL MAPS . 




| 3 - 


Fig. io.—Preliminary Map of Canal. Montana. 
Scale 4000 ft. to 1 inch. Contour interval 4 ft. 


















SURVEYS FOE EE SEE FOIES. 


57 


being a mile on a side. The original survey was made on 
a scale of 3000 feet to the inch* with a contour interval of 4 
feet. The plane-table was accompanied by a spirit-level to 
determine grade, in order that the canal line might be given 
the required fall per mile. 

24. Surveys for Reservoirs. —In making surveys of 
reservoirs for storage of water for city water-supply or for 
irrigation and similar purposes, the scale and contour interval 
depend necessarily on the dimensions of the reservoir. The 
former should be from 400 to 1000 feet to the inch, and the 
latter from 2 to 5 feet vertical interval. Special surveys 
should be made of possible sites for dams and waste-weirs on 
larger scales and with a contour interval of 1 or 2 feet, and 
several cross-sections of the dam site should be run and the 
topography taken in detail for a sufficient distance above and 
below the center line. If sufficient borings or trial-pits are 
sunk, a contour map of the foundation material may be 
constructed. 

Perhaps the most satisfactory manner of making surveys 
of reservoir sites is instanced in the following practical ex¬ 
ample of one made by the author. A standard or base 
transit (Art. 87) and level (Art. 129) line is first run across 
the dam site, carrying the same a little above the highest 
possible flow line of the reservoir. From this should start a 
main transit and level line which should follow up the lowest 
or drainage line of the reservoir basin (Fig. 11, A, D , G), 
and this should be extended until it reaches an elevation 
corresponding to that of the highest probable flow line of the 
dam. Bench-marks (Art. 135) should be left as this line 
progresses, and stadia distances measured (Art. 102), and level 
elevations taken to points within the range of the level-tele¬ 
scope. as at A, B, etc. Based on this main transit and level 
line, a plane-table and stadia line (Art. 101), accompanied by 
spirit-leveling, should be run from the highest flow line of 
the dam cross-section around the corresponding contour line 


5.8 SURVEYING FOR DETAILED OR SPECIAL MAPS. 

on ope side of the reservoir, H , I, J, etc., and if the land 
be clear, stadia and level sights may be taken to the other 
contour : lines within the range of the instrument, including 
sights on lines of equal elevation on the opposite side of the 
reservoir if the latter be small. If large, however, a number 
of flags may be located on the opposite side by triangulation 
(Art.. 73) or by stadia observations, and cross-section lines be 
run to these, from which the data for constructing a contour 
topographic map can be obtained as at / and L. 

Another example of a reservoir survey is illustrated in 
Fig. 12, which is a portion of the map of the Jerome Park 
reservoir site in the city of New York, and was platted on a 
scale of 400 feet to the inch with a contour interval of 10 feet. 
From such a map it is possible to compute the contents of a 
reservoir for each additional five feet of elevation, and on it 
land lines and property lines are shown in such manner as to 
indicate the damage which will be done by submergence. 

25. Survey of Dam Site.—A typical illustration of the 
topographic map resulting from the survey of a site for a 
dam for closing a storage reservoir is shown in Fig. 13. 
This survey was executed with a plane-table (Art. 73), chain 
(Art. 99), and spirit-level (Art. 129) on a field scale of 400 
feet to 1 inch, with a contour interval of 2 feet. The result 
of such a topographic survey is to indicate clearly the best 
alignment for the dam, providing the borings which must 
necessarily follow the selection of such alignment prove its 
feasibility. 

An example of a topographic survey executed for selec¬ 
tion of a site for a weir or diversion dam in a river is. that 
illustrated in PL III. This shows the topography of the flood- 
bed of the Snake River between its high bluff banks, as 
well as the contouring of the bed of the river as shown by 
soundings. On this is indicated the best alignment for the 
diversion weir as well as for the canal head and headworks. 
The field work of the survey was executed with transit. 




47X0 


SITE OF HEADWORDS 


SNAKE 


RIVER AND POCATELLO CANAI 


LINE S 


DETAIL OF FIELD SURVEYS 


A. D. FOOTE , ENGINEER 


A. J. WTLEY. ASST. ENGINEER 


1«89 


Plate II.—Contour Topographic Survey of Site for Diversion Dam, Snake River, Idaho. 












































5 UR VE V OF DAM SITE, 


59 



Fig. ii.—Contour Survey of a Reservoir Site. Montana. 






































60 SURVEYING FOR DETAILED OR SPECIAL MAPS . 



Fig. 1.2.—Portion of the Jerome Park Reservoir Survey. New York. 
Scale 400 ft. to 1 inch. Contour interval 10 ft. 


J 































SURVEY OF DAM SITE. 



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O 

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C/l 

w 

i-5 

Z 

►H 

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H 

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o 

w 

hJ 

tc, 

o 

ftC 

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Q 

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62 SURVEYING FOR DETAILED OR SPECIAL MARS. 

plane-table, chain, stadia, and spirit-level on a scale of 200 feet 
to 1 inch, with a contour interval of 2 feet. 

26. City Surveys.—Topographic surveys of cities are 
almost invariably made in conjunction with complete cadastral 
surveys of the same, and usually under three conditions: 

(1) In laying out a plan for city extension or in making a 
plan for a projected city where little or no construction of 
streets, etc., exists; 

(2) In making a complete survey of a city on which to 
plan future public works of all kinds; and 

(3) A topographic survey of a city may be made merely 
for the sake of obtaining the resulting map. 

An example of the first class of city topographic survey is 
that for the survey of the town site of the city of Allessandro, 
in southern California. Surveys of a town site on a compar¬ 
atively level tract were made by Mr. J. B. Lippincott with 
such detail that irregular streets, parks, and other city im¬ 
provements were planned on the resulting topographic map. 
The eastern portion of the tract mapped had a general rise of 
about one foot in one hundred, the roughness increasing 
toward the west until broken country was reached. The 
contour interval was one foot, and the scale 100 feet to the 
inch. A base line (Chap. XXI) was projected through the 
centre of the tract, measured with care, and stakes were set 
at every 500 feet. At each 2000 feet a right angle was 
turned off and lines run north and south to the boundaries, a 
large stake being set every 500 feet on these lines. After 
these were located, two transits were used, one on the base 
line and the other on the 2000-foot line, and the remaining 
stakes were located by intersection at the corners of the 500- 
foot blocks. Flags were placed on each iooo-foot stake for 
witness-stakes and to orient the plane-table, and levels were 
run from the center base line and around the outside of the 
tract, readings being taken carefully on all hubs. 

In making the topographic map a plane-table was used 


CITY SURVEYS. 


63 


(Art. 53) and a spirit-level (Art. 130) was set up near it, the 
levelman placing two rodmen along distant contours, though 
it was sometimes found that four rodmen could be used on 
various contours. The plane-table was set over a stake and 
oriented by one of the flags, and on it were plotted the posi¬ 
tions of the rodmen by stadia distances (Art. 102), and thus 
the contours were sketched. Sights were taken in the more 
level portions of the country, not closer than 50 or farther 
than 100 feet apart, and the rodmen were placed on a contour 
and kept upon it until they reached a distance of 500 or 600 
feet from the plane-table. By this means about one square 
mile was mapped in a week of fair weather. Where the slope 
increased toward the west, 2, 4, or 5 feet contours only were 
located, and the others interpolated by sketching. Forty- 
seven working days were employed in mapping 3.25 square 
miles, during which 25,400 points were located, or 7800 to 
the mile, the cost being about $300 per square mile. 

Of the second class of city surveys the two most promi¬ 
nent examples are those of the cities of St. Louis and Balti¬ 
more (Art. 27). The topographic survey of the city of 
Washington, made by the U. S. Coast and Geodetic Survey, 
is an example of the third and unusual class of city survey. 
This was made on a scale of 1 : 4800, and covers an area of 48 
square miles. The result is a topographic map pure and 
simple , unaccompanied by the placing of permanent monu¬ 
ments or the obtaining and recording of accurate measures, as 
is necessary in making a cadastral survey of a city. This 
survey was based on a minute triangulation (Chap. XXV) 
while all the details of the topography were obtained by means 
of the plane-table (Art. 53), stadia (Art. 101), and Y level 
(Art. 129). The contour interval was 5 feet, and these contours 
were based on lines of Y-leveling run along all roads, avenues, 
and railroads. The plane-table stations were placed so close 
together as to encompass within the distance from station to 
station all the area within the range of the stadia. No system 


64 SURVEYING FOR DETAILED OR SPECIAL MAPS. 


of precise bench-marks was left in the course of the leveling, 
but the Y level was freely used in tracing successive contours 
upon the ground, the points upon each contour being located 
by stadia. The degree of refinement of this survey seems 
excessive in view of the scale of the map, as the errors of 
actual location of the contours upon the map would greatly 
exceed the actual errors of leveling; moreover, as no provis¬ 
ion was made for continual revision of the maps by leaving 
easily recognizable monuments, the value of such a survey is 
limited by the many changes due to rapid suburban develop¬ 
ment, which would render such maps out of date within a 
very short period of time. 

27. Cadastral and Topographic City Survey. —The 

topographic surveys of the cities of Baltimore and St. Louis 
were made in conjunction with complete cadastral surveys , and 
show all property lines, widths from building line to building 
line in all streets, dimensions, and other incidental data rela¬ 
tive to buildings, both public and private. The most com¬ 
plete example of such a survey is that furnished by the city of 
Baltimore. This was based on a system of triangulation ex¬ 
ecuted with precision (Chap. XXV) and connected with a 
base line measured with much care with a 300-foot steel tape 
(Chap. XXI). This triangulation covers 54.7 square miles, 
and for its execution required several high observation-towers 
in addition to existing structures. Between the located tri- 

1 

angulation points was an adjusted system of steel-tape 
traverse lines (Art. 87), executed in such number that no 
closed circuit of traverse exceeded 7500 feet in length. By 
these traverse lines there were located 3740 stations. Pre¬ 
cise levels (Art. 140) were run over an area of 29.51 square 
miles. These levels included 141 miles of duplicate line, in 
which were established 606 permanent bench-marks, while 
elevations were taken at every street intersection by ordinary 
Y levels (Art. 129). The primary control averaged three 
triangulation stations and forty-two traverse stations per 


CADASTRAL AND TOPOGRAPHIC CITY SURVEY. 65 


square mile in the unbuilt sections of the city. On this 
control there was constructed a detailed topographic map on 
a scale of 1 : 2400 and with contours of 5 feet vertical inter¬ 
val (Fig. 14). In the execution of this work there were run 
many miles of stadia traverse and Y levels and of taped 
measurements of street widths and building-line dimensions, 
etc. 

Three methods of obtaining topography were adopted: 
(1) that by transit and stadia, accompanied by notes worked up 
and plotted in the office ; (2) that by means of plane-table and 
stadia, with complete map made in the field on the plane-table ; 
and (3) that by transit and stadia, with notes worked up and 
plotted on a crude drawing-board in the field. The first two 
methods were employed in surveying only small areas, and 
were abandoned successively as not satisfactory. The third 
was that which was ultimately employed in mapping the 
larger portion of the city. In the prosecution of this latter 
method of work high-grade transits with fixed stadia wires 
and vertical and horizontal circles reading with verniers to 30 
seconds were employed. All notes, as rapidly as obtained by 
measurement and by angulation, were plotted with an 8 
inch protractor (Art. 89) and boxwood scale on the field 
drawing-board. Previously there had been plotted on the 
field sheets the primary triangulation and primary traverse 
locations (Chaps. XXV and XXIII) with lengths and azi¬ 
muths of lines joining stations, and elevations of precise- 
level bench-marks. The party organization consisted of a 
topographer, a recorder, a draftsman, a levelman, and two 
stadiamen. As rapidly as the topographer read azimuths, 
distances, and vertical angles, the draftsman plotted the same, 
and the recorder worked out elevations furnished by the 
topographer and the levelman. After all observations had 
been taken and the horizontal locations and elevations plotted, 
the contours were drawn in on the field board by the topog¬ 
rapher, and the party moved to the next station. The total 


66 SURVEYING FOR RETAILED OR SPECIAL MAPS . 



Fig. 14* Topographic and Cadastral Map of Baltimore, Md. 
Scale 200 ft, to i inch. Contour Interval 5 ft. 























































































































































































COST OF LARGE-SCALE TOPOGRAPHIC SURVEYS. 67 


area of topographic survey was 32.2 square miles, in which 
there were located 213 miles of streets and alleys, 1147 pre- 
cise points were occupied, 2320 stadia stations occupied, and 
I34» 2 °9 sights were taken, the average being 10 per acre. 

28. Cost of Large-scale Topographic Surveys.— 
Special topographic surveys are usually prosecuted with a 
view to showing all the topographic details of a limited area, 
and are executed with such minuteness that the resulting map 
may be plotted on a large scale. Not uncommonly such sur¬ 
veys are of cadastral thoroughness, and the results may then 
be plotted on such a scale as will permit of showing in plan 
the minutest detail of houses and other structures. 

Such a survey is the British Ordnance survey, plotted on 
a scale of 1 : 2500, and the topographic and cadastral surveys 
of the cities of St. Louis and Baltimore (Art. 27). Also 
topographic surveys for railroads, reservoirs, etc., plotted 
usually on scales of 400 to 1000 feet to one inch and with 
2-foot to 10-foot contours (Arts. 22 to 24), on which property 
lines are also shown. 


Table L 

SCALE AND COST OF DETAILED TOPOGRAPHIC MAPS. 


Country. 

Scale. 

Relief. 

Cost per sq. mi. 

India . 

4 in. to 1 mi. 

Hachures 

$ 26.50 

BaHpn . 

t:5000 

1: 10,000 


80.OO 

U. S. Coast Survey. 

Contours 

212.00 

U. S. Lake Survey. 

1: 10,000 

i i 

120 .CO 

U. S. Miss, and Mo. 
River Com. 

1 : 10,000 

i 1 

51.00 

U. S. Coast and Geo¬ 
detic Sur. of Dist. of 
Col . 

1 : 4800 

5-ft. contours 

3,000.00 

Butte (Mont.) Special U. 
S. G. S . 

1 : 15,000 

K> 

0 

1 

*-K 

83.OO 

Perkiomen Watershed, 
Penn . 

1 : 4800 

IO-ft. 

i45-oo 

Croton Watershed, N. Y. 

3 in. to 1 mi. 

20 ft. 

17-50 

Connellsville Coke Re- 
gion, Penn . 

1 : 19,200 

IO-ft. 

116.36 






















CHAPTER IV. 


GEOGRAPHIC AND EXPLORATORY SURVEYS. 

29. Geographic Surveys. —The object of a geographic sur¬ 
vey is to fix the relative positions of points on the surface of 
the earth so that they can be referred accurately to a tangent 
plane and be therefore independent of the sphericity of the 
earth. The geographic survey of an extended area consists 
of three parts: 

1. A geodetic survey, which permits of the projection of 
a primary system of controlling points on such a tangent 
plane. 

2. Of a plane survey, for the projection of a system of 
intermediate controlling points upon the same plane and 
adjusted to the primary system. 

3. Of a hypsometric survey, for the determination of the 
distances of the points established by the other two sur¬ 
veys above or below an assumed datum or basal plane of 
elevation. 

The results of a geographic survey are presented— 

1. In a geographic map, which is intended to give as 
complete an image of the area surveyed as the scale of rep¬ 
resentation will permit; and 

2. In a report on the physical and statistical characteris¬ 
tics of the region surveyed. 

The methods employed in the field execution of the geo¬ 
graphic survey are described hereafter under various titles. 
An essential preliminary to the geographic survey is a ge¬ 
odetic survey based on astronomic positions, and the mode 

68 


INSTRUMENTAL METHODS EMPLOYED. 69 

of obtaining this fundamental information is explained in 
Parts V and VI. With such primary control as is furnished 
by the geodetic survey the details of the geographic survey 
are executed by some of the various methods explained in 
Chapter II and Part II. They are essentially similar to 
those employed in the making of topographic surveys, differ¬ 
ing therefrom chiefly in the employment of cruder and more 
rapid methods. Moreover, the amount of information to be 
gathered is more scattered and less detailed than that pro¬ 
cured by topographic surveys, because the scale of the 
resulting map is smaller and therefore will not permit of the 
representation of minor details. 

30. Instrumental Methods Employed in Geographic 
Surveys. —For the making of a geographic map the primary 
control must be executed by geodetic methods, but this need 
not be of the highest degree of accuracy, but only of such 
quality that the resulting errors will not be appreciable upon 
the scale of the map. For filling in the intermediate details 
the most useful instrument is the plane-table (Chap. VII), 
which may be employed for the execution of secondary and 
tertiary triangulation, for road traverses (Chaps. IX and X), 
and as a sketch-board on which to fill in the details of 
topography (Arts. 13, 15, and 17). 

In the course of such work the methods employed will be 
of a crude nature. Signals will rarely be erected, natural 
objects being sighted both in the triangulation and in the 
traverse, and the number of stations and locations will be 
relatively few and far apart one from the other. They must, 
however, be fixed with such accuracy upon the scale of map 
that there will be at least two or three located points to each 
square inch of map surface. Thus on a map scale of two 
miles to one inch there may be an average of less than one 
location per square mile. On a scale of four miles to one 
inch there may be but one location to every two square 
miles. Again, on the latter scale, plane-table stations would, 


7 o 


GEOGRAPHIC AND EXPLORATORY SURVEYS. 


under favorable circumstances, be placed at an average dis¬ 
tance of ten miles apart, and from each there would be 
sketched an approximate area of one hundred square miles. 
For it will be realized that on the scale of the map this 
implies sketching from each station to a distance from one to 
one and one-quarter inches in each direction and over terri¬ 
tory controlled by intermediate locations averaging one-half 
inch apart. 

The intermediate details of the geographic survey ex¬ 
ecuted for small-scale maps should be filled in by the ordi¬ 
nary traverse methods, performed, however, with instruments 
fitted only for the execution of approximate work. Thus a 
very light traverse table (Arts. 61 and 63) or a prismatic 
compass (Art. 91) should be used for directions, while distance 
may be obtained by wheel (Art. 98) or pacing (Art. 95). 
Elevations will be determined in the prosecution of a geo¬ 
graphic survey of this character by several methods, the 
amount of basal spirit-leveling (Chap. XV) being of the very 
least, perhaps only a few fundamental elevations per map 
sheet. The more important elevations will be obtained by 
trigonometric levels of primary or secondary quality (Chap. 
XVII), and the larger proportion of the intermediate eleva¬ 
tions will be obtained by mercurial barometer with aneroid 
for elevations of less moment (Arts. 1 70 and 174). 

31. Geographic Maps. —A geographic map is generally 
plotted on a small scale, corresponding with the least scales of 
general governmental surveys (Table II), and the limits of such 
scales are roughly between about one mile to one inch and six 
miles to one inch. For larger scales the resulting map might 
be classed as topographic, and for smaller as exploratory. On 
geographic maps various conventional signs (Art. 195) are 
employed to represent hydrography or drainage, culture or 
works of man, and relief or surface undulations. Such drain¬ 
age features as streams, lakes, and ocean margins as may be 
of sufficient size to permit representation on the scales selected 


GEOGRA PH1C AIA PS . 


71 

are shown, Such cultural features as are of a strictly public 
nature, as railways, the more important highways, cities, and 
political boundaries, should be shown. Surface undulations 
should be clearly represented by one or other of the various 
conventions. According to the scale of the map and the 



Fig. 15.—Field Sketch Map made on Plane-table in Alaska. 

E. C. Barnard, Topographer. Scale 4 miles to 1 inch. 

quality of the field survey, such representation may be by 
contour lines of considerable vertical interval, by sketch or 
broken contours representing relative differences of relief, 
or by means of hachures which represent in a conventional 
way degrees of relief, absolute relief being shown only by 
written figures of elevation. 

In Fig. 15 is shown a portion of the sketch contour 
map made in Alaska by E. C. Barnard of the U. S. 
Geological Survey. In Fig. 16 the same map is shown 
after it has been drawn up in office. The contour interval 
of this is 200 feet, and the scale 4 miles to one inch. Yet 












7 2 


GEOGRAPHIC AND EXPLORATORY SURVEYS. 


these are not true contours, and should preferably not have 
been represented as such, since the amount of vertical con- 


Fig. 16.—Geographic Contour Map made from Fig. 15. 

Scale 4 miles to 1 inch. Contour interval 200 feet. 

trol was too small to give that exactness implied by contour 
lines. 

32. Features Shown on Geographic Maps. —A geo¬ 
graphic map should show with sufficient completeness all the 
important topographic features of the area surveyed. It should 
depict especially physiographic peculiarities, which are the key 
to the origin of topographic forms. It will thus be realized 
that in their execution the geographer should have a clear 
knowledge of the relations of geology to topography (Art. 45). 

Accordingly the amount of the instrumental control re¬ 
quired will be the minimum which will permit accurate repre¬ 
sentation of the essential and predominant features. Between 
this control the shape and positions of the various streams 
may be sketched in such a manner as not only to show their 
direction, but their changes of direction as determined by 


















GEOGRAPHIC REPORTS. 


73 


accidents of broken or displaced stratification or the slope of 
the surface over which they flow. Moreover the map will 
distinguish between the rounded slopes of a synclinal and the 
abrupt sides and angular sections of an anticlinal gorge. It 
will show at a glance the position of a fault in the stratifica¬ 
tion by precipitous slope and exposed strata on one side, and 
on the other the gentle declivity of tilted surface rock. It is 
thus evident that the geographer must be largely guided in 
his depiction of the terrane by his knowledge of the geologic 
structure, so that the resulting map, while well controlled in 
relative place by instrumental locations, will, because of the 
necessity of generalizing topographic forms imposed by the 
small scale employed (Chap. VI), bring out the essentials 
or keynotes of such form rather than permit their burial 
under a mass of detail which is not essential to the purpose 
of the map. 

33. Geographic Reports.—The smaller-scale geographic 
maps executed by governments, and in a few instances by 
railway enterprises in connection with surveys made for the 
gathering of general information relative to unexplored 
regions, should show not only the topography of the region 
surveyed, but the outlines of its forested areas. Above all it 
should be accompanied by reports on all those scientific and 
economic facts which will aid in developing the region under 
examination. Examples of such surveys are those executed 
by the Hayden survey in Colorado and Wyoming, and by 
the Wheeler survey in various portions of the United States 
west of the 100th meridian. 

In both the resulting maps are based on geodetic con¬ 
trol, and are published on various scales according to the 
objects of the survey. In the case of the Hayden survey a 
general scale of four miles to one inch was adopted (Fig. 20), 
and differences of elevation were shown approximately by 
contours having an interval of 200 feet. In the case of 
the Wheeler survey two general scales of four (Fig. 19) 


74 


GEOGRAPHIC AND EXPLORATORY SDR YE VS. 


and eight miles to the inch were used in various localities, 
and the surface relief was depicted by hachures accompanied 
by occasional figures of elevation, actual elevations not being 
shown, as in the case of the Hayden survey, by contour lines. 
In the case of both surveys, in addition to showing the cul¬ 
ture, drainage, and relief, the general topographic maps are 
accompanied by special maps showing the distribution of 
timbered, pasture, and barren land, and by other maps show¬ 
ing the surface geology. Finally, the whole was accompanied 
by extensive printed reports detailing the important scientific 
and economic features of the regions examined. 

34. Scale and Cost of Government Geographic Surveys. 
—All civilized nations appreciate the value and necessity of 
good topographic maps of their territory. The principal 
nations of Europe have completed surveys that will generally 
subserve the purposes of geographic maps, or are now en¬ 
gaged upon such work. These European surveys are all 
based upon a computed triangulation and are usually made 
upon a scale not far from one mile to one inch or 1 : 63,360. 
They are sometimes larger and sometimes smaller. Their 
scales range between one mile for Great Britain up through 
Austria, France, Norway, Germany, and Russia, to two miles 
in the latter country. And from Great Britain they range 
down with larger scales through Sweden, Italy, Spain, Den¬ 
mark, and Switzerland, the scale for the latter being a little 
larger than two inches to one mile. 

A study of these maps is of value in determining the 
scale which should be adopted for a general geographic map 
of the United States, and as a result the scales chosen for the 
latter are from one to two miles to one inch. It is believed 
that the larger scale offers the best opportunity for the ex¬ 
pression of such features of the country as the engineer, 
legislator, or investor desires to see expressed with some de¬ 
tail on a general map. If a still larger scale map is desired, 
it is usually for a small area, and for this purpose the indi- 


SCALE AND COST. 


7 5 


Table II. 

SCALE, COST, AND RELIEF OF GOVERNMENT GEOGRAPHIC 

MAPS. 


Country. 

| 

Scale. 

Relief. 

Cost 

per sq. mi. 

India. 

I mile to i inch 

Hachures 

$ II. OO 

Austria. 

1:75,000 

Hachures and 
contours 

400.00 

Baden. 

1125,000 

Contour 

22.20 

Belgium. 

1:20,000 
1:40,000 

One meter 
Contours 

167.OO 

France. 

1:10,000 

1:20,000 

Hachures 

52.00 

Great Britain. 

1 mile to 1 inch 

Hachures and 
contours 

184.OO 

Italy . 

1:100,000 

Contours, 5 and 
10 meters 

30.00 to 

45-00 

Prussia. 

United States Geological 
Survey : Middle Atlantic 

1 100,000 

Con tours, 5 meters 

71.00 

and Eastern States. 

Geological Survey : South- 

1:62,500 

Contours, 20 ft. 

10.00 

ern and Western States. 
Geological Survey, West- 

1:125,000 

Contours, 100 ft. 

4.00 

ern States. 

1:250 000 

Contours, 200 ft. 

1.75 

Havden Survey. 

4 miles to 1 inch 

Contours, 200 ft. 

2 . IO 

Wheeler Survey. 

4 4 4 4 

4 miles to 1 inch 
All; 2 to 8 miles 

Hachures 

2.25 


to 1 inch 

< 4 

1.50 


vidual desiring the map should, and probably would, make 
his own special surveys, as they would be conducted with a 
view to the inauguration of active engineering operations. 
The contour interval chosen for the geographic map of the 
United States varies according to the topography and hori¬ 
zontal scale from five to one hundred feet vertically. The 
smaller contour intervals are employed especially on very 
level costal plains, while the larger intervals must be used for 
the expression on the same scale of steep mountain slopes 
and valley walls. It has been found that this range of con¬ 
tour interval gives the best mean value for the expression of 
all characters of topographic form, permitting the proper de¬ 
piction on the scale chosen of the steepest mountains and yet 



































76 

0 


GEOGRAPHIC AND EXPLORATORY SURVEYS. 


giving a fair idea of the value of the slopes on the more level 
surfaces. 

35. Exploratory Surveys. —One of the essentials of an 
exploration is some form of survey which shall record the 
appearance of the country traversed. The primary requisite 
in such a survey is some means of measuring directions and 
distances along routes of travel. A well-equipped expedition 
should be provided with several varieties of instruments for 
this purpose lest some be lost or injured, and in order that 
those best suited to the exigencies of the case may be 
employed. Sometimes no effort is made to fix the geo¬ 
graphic position of such surveys, but ordinarily and where 
the work is conducted under scientific auspices means are 
provided for the determination of latitude, longitude, and 
azimuth by simple instruments and with approximate accuracy. 

Azimuths may be measured in route surveys with pris¬ 
matic compass, or by means of a light plane-table, or with a 
light transit (Arts. 91, 63, and 85). 

Distances may be measured by stadia, by pacing, by timing 
the gait of animals or of a boat rowed or drifting, and in 
extreme cases by mere eye estimation (Arts. 102, 95, and 96). 

Where the exploration is of a compact area rather than of 
a route the survey may be best executed by trigonometric 
methods (Chap. IX), with light plane-table, with transit, or 
by photo-surveying methods (Chap. XIV). In such a case 
elevations may be conveniently determined by vertical 
angulation (Art. 160), when the resulting map will be rather 
geographic than exploratory in quality. 

Astronomic position is determined in such a survey by 
latitudes observed with sextant, and longitudes obtained by 
chronometer (Arts. 336 and 328), or by lunar photographs 
(Chap. XXXVII). Azimuths are readily obtainable by obser¬ 
vations on polaris with theodolite (Art. 312). In a compact 
trigonometric survey several careful determinations of lati¬ 
tude, longitude, and azimuth made at one point only are 


COMPARISON OF SURVEYS. 


77 


necessary. In running a route survey latitudes should be 
observed at distances not exceeding fifty miles, longitudes 
as frequently as convenient, according to the method, and 
azimuths on nearly every clear night. 

The sources of error in such a crude route traverse are in 
the measurement of directions and distances. The former 
will be but slightly in error for the small scale of map se¬ 
lected if frequent azimuths are observed. Errors in dis¬ 
tance will be fairly well compensated by the observations 
made for latitude. The most satisfactory way of determining 
longitude under such conditions is by means of ships’ chro¬ 
nometers read at the point at which the expedition starts out, 
provided that be on the seacoast (Art. 328). The plotting 
of the final map will aid in keeping the longitude fairly well 
in check. 

Elevations should be recorded by means of aneroid ba¬ 
rometer (Art. 174), and the eccentricities of this may be kept 
in check by carrying a cistern mercurial barometer, which 
should be read hourly at each camp (Art. 170). Where 
the circumstances permit, a base barometric station should be 
established, at which the moving barometer should be com¬ 
pared with the stationary standard, and the latter should be 
read hourly throughout the duration of the expedition in or¬ 
der to permit of a reduction of the synchronous observations 
of the moving barometer (Art. 169). 

36. Exploratory and Geographic Surveys Compared. 
The following is an interesting comparative group of maps 
illustrating the result of surveys of various degrees of accu¬ 
racy. Fig. 17 is a small portion of the sketch map accom¬ 
panying the report of Captain Zebulon M. Pike, and made in 
1807. This includes the headwaters of the Platte and Ar¬ 
kansas rivers in Colorado, the point marked “ Highest peak” 
being the summit now known as Pike’s Peak, and “ Block¬ 
house ” being presumably the present location of Canyon 
City. This sketch map was made without the aid of instru- 


78 GEOGRAPHIC AND EXPLORATORY SURVEYS. 



l'eUow Stone B . 
of the Missour 


The head waters of this interlock 
with the main S.R. branch of 
I Platte R. which issues from 
\ the Mountains 


La Platte 


Highest Peak 




merits, and is entirely uncontrolled in distance or direction 
other than by estimates. 

» 361 _351_34°|_33]_ 4 o° 


361 351 341 33T 

Fig. 17—Capt. Zebulon Pike’s Map about Pike’s Peak, Colo. 


1S07. 


Fig. 18 is a small portion of a map published in the re¬ 
port of Captain J. C. Fremont of an exploration across the 
Rocky Mountains in 1845. Geographic position is ap¬ 
proximately fixed by means of sextant observations for lati¬ 
tude, chronometer, and lunar observations for longitude, and 
barometric observations for height. Between these sparsely 
scattered astronomic positions directions and distances are 
by estimate only, the route, however, being sketched at the 
time of travel. 

bigs. 19 and 20 cover small portions of the same area 
on the west slope of Pike’s Peak. The first is a portion of 










COMPARISON OF SURVEYS. 


79 


the U. S. Engineers’ geographic map, scale of four miles to one 
inch, and made between 1873 and 1876. This map shows 
relative relief by means of hachures, actual relief being shown 
-only by figures of elevation, the result of barometric or 
trigonometric observations. The surveying was executed by 
means of secondary triangulation with transit, expanded from 



Fig. 18. —Capt. J. C. Fremont’s Map about Pike’s Peak, Colo. 1845. 

a primary triangulation executed with theodolite and based on 
an astronomic station and carefully measured base line. In¬ 
termediate details were sketched in from the secondary trian¬ 
gulation stations and by odometer traverses of roads. Fig. 
20 is a small portion of the Hayden map covering the same 
area. This is a geographic map also on a scale of four miles 
to the one and executed at about the same time as the U. S. 
Engineers’ map. The method of survey was practically the 















8o 


GEOGRAPHIC AND EXPLORATORY SURVEYS, 



Fig. 19. —Wheeler Map about Pike’s Peak, Colo. 1876. 

Scale 4 miles to 1 inch. 






Fig. 20.—Hayden Map about Pike’s Peak, Colo. 1875. 
Scale 4 miles to 1 inch. Contour interval 200 feet. 














82 


GEOGRAPHIC AND EXPLORATORY SURVEYS. 


same, but many more elevations were determined both by 
barometric and trigonometric methods, and from these approxi¬ 
mate contours of two hundred feet interval were sketched, 
thus giving the relief with greater relative accuracy. 

In Figs. 21 and 22 are shown small portions of the same 
area as mapped by the U. S. Geological Survey, the first in 1892 
and the second in 1894. Fig. 2 1 is a fragment of an accurate 
geographic map on a scale of two miles to one inch, and with 
differences of elevation represented by contours of one hun¬ 
dred feet interval. Field-work was based on a careful primary 
triangulation and was executed by means of plane-table with 
telescopic and sight alidade for direction and vertical angula¬ 
tion, odometer traverses of roads, and sufficient spirit-leveling 
and stadia work to fill in the details. Fig. 22 is a large- 
scale topographic map of the same area executed on a scale of 
1 : 25,000, approximately two and one-half inches to one mile, 
and with a contour interval of fifty feet. This was based on 
a plane-table triangulation and spirit-leveling accompanied by 
stadia traverses and intersections for both vertical and hori- 
zontal detail. 

37. Methods and Examples of Exploratory Surveys. 

—The field-work of exploratory surveying includes the making 
of some form of record of the geography of the country 
passed over, which may be either kept in note-books and 
worked up in office or may be drafted in the field upon a sketch 
plane-table. Such surveys may be of a route only, especially 
where the course traveled is the bed of a narrowly confined 
stream, or through woods when little can be seen of the sur¬ 
rounding country, or it may be of an area when the explorers 
are traversing open country or high ridges which permit of an 
extended outlook over the region surrounding them. 

The personnel of an exploring party should consist, if pos¬ 
sible, of one individual qualified to conduct any form of sur¬ 
vey, be it by transit, plane-table, compass, stadia, or estimate, 
as the circumstances may demand, and also competent to 


Portion of U. S. Geological Survey Map about Pike’s Peak, Colo. 1892. 
Scale 1 to 125,000. Contour interval 100 ft. 










































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EXAMPLES OF EXPLORATORY SURVEYS. 


8 7 


determine astronomic position by transit, sextant, and similar 
instruments. In addition, at least one other member of the 
party should be versed in the sciences of geology and biology 
in order that he may understand how to collect information of 
the mineral resources, the flora and fauna of the region trav¬ 
ersed. A photographic camera should be carried for record of 
the aspect of the landscape or of details seen. The results of 
the work of this member of the party will accompany the map 
in the form of an illustrated report. 

The instrumental equipment of such a party should include 
various forms of surveying instruments, at least two or three 
methods of measuring directions and distances being dupli¬ 
cated lest any of the instruments be lost or destroyed. There 
should also be carried aneroids and mercurial barometers for 
the determination of heights (Arts. 174 and 170). The 
instruments for determining direction should include, if pos¬ 
sible, a light mountain transit (Art. 85) specially provided 
with prismatic eyepiece and striding-level for the determina¬ 
tion of latitude and azimuth ; a sextant for astronomic obser¬ 
vations and the measurement of horizontal and vertical angles 
(Art. 336); a 1 ight plane-table with sight alidade (Arts. 56 
and 62), to supersede the transit in rougher surveys when 
necessary, and a prismatic compass (Art. 91) to replace either 
of the above. Distances may be measured by means of the 
stadia hairs in the transit, a light pole being marked with 
bands of white cloth or string or other device as a stadia-rod 
(Arts. 101 and 112). Distances will be obtained in addition 
by triangulation (Chap. IX) or pacing, or by means of the 
linen tape, or by time estimate on land or in floating down 
streams (Arts. 65, 97, and 96). 

Two examples of exploratory surveys are illustrated in 
Figs. 23, 24, and 25. The first is that of a route trav¬ 
erse made in Alaska in 1898 by Mr. W. J. Peters of the 
United States Geological Survey. This is the first authentic 
survey of that region and was made with plane-table and ali- 


88 


GEOGRAPHIC AND EXPLORATORY SURVEY \ 



W. J. Peters, Topographer. Scale i to iSo.ooo. 




















EXA MPL ES OF FXPL OKA 1 'OK V S UK VE VS. 



8 a 


Fig. 24.—Exploratory Route Survey, Alaska. Final Drawing. 
Scale 10 miles to 1 inch. Sketched contour interval 100 feet. 






9 o 


GEOGRAPHIC AND EXPLORATORY SDR PE VS. 



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EXAMPLES OF EXPLORATORY SURVEYS . 


9 1 


dade for directions, transit for latitude, and distances by stadia 
and pacing on the portages, and time and eye estimate on the 
water. It is complete in that it shows by sketch contours 
the shapes of the surrounding hills and gives from hearsay and 
other sources some of the detail of the region included within 
the route of exploration. Fig. 24 is the final office drawing 
of the same. 

Fig. 25 is an exploratory map, not of a route, but of a 
compact area resulting from surveys by Mr. Willard D. John¬ 
son of the U. S. Geological Survey for the Bureau of American 
Ethnology in the year 1895. This survey was made with a 
light traverse plane-table, oriented by compass and alidade. 
It was executed by plane-table triangulation, started at the 
international boundary line, and the map is dependent thereon, 
but is checked by the coast line of the U. S. Hydrographic 
Survey. The area mapped covers 10,000 square miles. The 
geographic position of the map is dependent upon the work 
of the boundary and coast-line surveys. This is a beautiful 
example of a well-executed sketch map by an expert topog¬ 
rapher, although but sparsely controlled by instrumental 
locations. 


CHAPTER V. 


MILITARY AND CADASTRAL SURVEYS. 

38. Military Surveys.—Ordinary maps are sufficient to 
enable one to follow the operations of a campaign, but for 
planning military operations detailed topographic maps are 
essential, because the merest trail or smallest stream or valley 
or undulation of the ground may for a time become of the 
greatest importance either for offensive or defensive pur¬ 
poses. 

Topography for such uses, however, calls for the most 
simple problems of mathematical surveying, rarely has recourse 
to plane trigonometry, and never employs the principles of 
spherical trigonometry, because the areas included within 
any separate map sheet are small. Military topography is in 
fact the art of obtaining a detailed representation of but a 
moderate extent of the earth’s surface. The resulting map 
should exhibit the important lines and characteristic objects 
on the ground, not only including main streams, railways, and 
mountains, but smaller watercourses, roads, and foot-paths, 
houses, enclosures, ditches, excavations, embankments, fences, 
hedges, walls, and the minor undulations of the surface, 
especially abrupt ledges. 

The preparation of the military map consists of two oper¬ 
ations : 

I. The projection in proper relative position on a plane 
surface of the main outlines of the country; and 


92 


MIL / 7'A R Y S UR VE YS. 93 

2. Of leveling, by means of which may be represented the 
slopes, elevations, and depressions of the ground. 

In addition there must be prepared a memoir including 
essential information which it is impossible to exhibit in 
graphic form, such as kind of road, its surface and state of 
repair, descriptions of bridges, character of their approaches, 
depth and rapidity of current in watercourses, nature of bot¬ 
tom, statistics of number of inhabitants, supply of provisions 
and animals, etc. 

The basal outline map upon which the military topography 
is to be exhibited may be a good topographic map, which 
should ordinarily be constructed in a manner similar to that 
described for the making of small-scale topographic maps 
(Chap. II). Having such abase, it is then possible to enlarge 
it to a sufficient scale to permit of representing upon it those 
details of information which are essential to military maps, 
and these may ordinarily be obtained by less accurate 
methods of survey, by the use of the plane-table or cavalry 
sketch-board (Arts. 57 and 64), supplemented by odometer, 
stadia, range-finder, or by pacing or by counting the paces of 
a horse (Arts. 98, 102, 117, and 95). With such instruments 
it becomes possible to sketch on the base map with some 
accuracy the positions of the hedges, minor watercourses, 
etc., and to enter in a note-book the data which go to make 
up the memoir. 

That form of surveying which produces a military map 
may be classed as irregular surveying , and consists ordinarily 
of rapid, interrupted journeys having for their object the repre¬ 
sentation of the natural and artificial features of the country 
with the maximum exactitude consistent with the greatest 
rapidity of execution, and it is therefore evident that they are 
based upon the same principle as are more elaborate surveys 
(Art. 9). The differences between them consist chiefly in the 
use of more portable and less bulky instruments, in the substi- 


MILITARY AND CADASTRAL SURVEYS . 



Fig. 26. —Skeleton of Route from Best Available Map. 
After Capt. Willoughby Verner. Scale 1^ inches to I mile. 




MILITARY RECONNAISSANCE WITH GUIDE MAP. 95 


tution of pacing or range-finding for the chain, and often in the 
estimation of distances and details by the eye. Such surveys 
are commenced by the determination of principal points by 
triangulation (Chap. IX) if such does not already exist, or by 
identifying triangulation points already existing. To these 
triangulation points further details are referred, and further 
irregular surveys are planned by a general glance at the field 
from them. Vertical measurements, which are essential in 
the representation of surface slopes, are but relative, and may 
be best had by the use of the aneroid. 

39. Military Reconnaissance with Guide Map. —The 
following examples from Captain Willoughby Verner show 
the mode of converting a small-scale geographic map into a 
detailed military map. In Fig. 26 is shown the outline of 
the route of the proposed reconnaissance. This is an en¬ 
larged copy of road and drainage crossings taken from a one- 
mile British ordnance map. In the following figure (No. 27) 
is shown the mode of filling in important military informa¬ 
tion on such a base, the notes all being made on the map in 
the course of a quick cavalry ride. In Fig. 28 is shown 
the final drawing made in camp from the notes taken on the 
preceding cavalry sketch map. All of this information was 
obtained without the use of instruments other than a sketch- 
board carried on the wrist of the topographer. 

40. Military Reconnaissance without Guide Map— 
In Fig. 29, taken from Willoughby Verner, is shown a 
portion of a river reconnaissance on the Nile executed from a 
river steamer. The distances were reckoned by time, and the 
directions by magnetic bearings. The important military 
information accompanying this is in the form of marginal 
notes. 

An extended reconnaissance sketch was made in the 
Soudan by Captain Verner, with the range-finder for distances, 
and light plane-table board for directions. This was more than 


96 


MILITARY AND CADASTRAL SURVEYS. 



% 


Fig. 27. —Sketch Route of Fig. 26 Filled in with Field Notes, 

After Capt. Willoughby Verner. 

















MILITARY RECONNAISSANCE WITH GUIDE MAP. 97 



Fig. 28.— Sketch Route of Fig. 26 Filled Out from Field Notes of 

Fig. 27. 

After Capt. Willoughby Verner. 





















9 8 


MILITARY AND CADASTRAL SURVEYS , 



Fig. 29. —Reconnaissance on Nile River from Gordon’s Steamer. 

Distance by- Time 7 . 


After Capt. Willoughby Verner. 





































MILITARY RECONNAISSANCE WITHOUT MAP , 


99 



LOFC. 


After Capt. Willoughby Verner. 















































IOO 


MILITARY AND CADASTRAL SURVEYS. 


a route reconnaissance, the territory covered being developed 
by plane-table triangulation (Chap. IX) and range-finder (Art. 
116), so as to cover a fairly extended area of the country. 
Such reconnaissances need not be accompanied by memoir, 
since all necessary notes are placed upon the margin of 
the map. They should, however, when made to develop 
the position of an enemy, be accompanied by perspective 
sketches similar to that illustrated in Fig. 30. 

41. Detailed Military Map. —In many of the more im¬ 
portant operations of the Civil War, such as the final action 
about South Mountain, time was afforded for the making of 
careful topographic surveys for the procurement of military 
information. Advantage was taken of such opportunity 
by making detailed surveys, in which roads were traversed 
with transit and chain or odometer (Arts. 87, 99, and 98), 
elevations measured with level as well as aneroid (Arts. 129 
and 174), and the resulting map gave all the detail of relief 
and cultivation which could be of use to the commanders. 
Fig. 31, reproduced from one of these maps, is an excellent 
example of the character of detail procured in making such 
surveys. 

Where a number of persons are employed' in surveying a 
region not previously mapped, as that described, accurate 
results can be obtained only by extension of a brief skeleton 
triangulation, locating a number of conspicuous points, or by 
running a few very careful transit and chain traverses of im¬ 
portant roads. This laid down on a large sheet is the basis 
of the survey, securing for it accuracy independent of the 
minor errors which must pervade the more detailed surveys 
executed with inferior instruments and in haste. Such a 
survey can only be made when there is ample time and pro¬ 
tection and suitable instruments are available. Moreover, 
such survey would only be necessary in unknown country. 
Where pressed by lack of time or protection only the crudest 


MILITARY SIEGE MATS 


IOI 


sketching implements and methods can be employed, and the 
survey must be a rapidly executed sketch accompanied by 



Fig. 31.—Military Map of Operations about South Mountain. 

memoranda of the country immediately adjoining the line of 
march. 

42. Military Siege Maps. —In siege operations sketch 
maps, photographs, and, when possible, more accurate sur¬ 
veys based on instrumental observations; are made of the po- 























102 


MILITARY AND CADASTRAL SURVEYS . 


sition attacked. These, so far as possible, are executed in such 
detail as to show not only the immediate surroundings with a 
view to their availability for the purposes of the attack, but 
especially the details of fortified places in order to develop 



SKETCH OF 


FORT DEFENSES 

MADE DURING THE SIEGE 
By Lieut, J.A.CHALARON, •* 
5th Co. Wash. Artillery * 
Comdg. at BATTERY NO. 3 
■ Confederate 
V- Union 


#.*** 


MORTARS 

■Hill 


8-GUN BATTERY, 4 RIFLED 
20-PDR. PARROTTS 
AND 4 RIFLED 10 PDR. * 
PARROTTS * 


Fig. 32.—Military Siege Map. 


their weak points and their strength. Fig. 32, reproduced 
from maps accompanying the Records of the War of the Re¬ 
bellion, is a map of this sort executed during the siege of 
Spanish Fort. 

43. Military Sketches and Memoirs. —Information must 
be procured from the inhabitants, spies, or other sources, and 
the military map filled out as well as may be from verbal 
descriptions. Itineraries of routes should be plotted and kept 
in memoir form for the guidance of bodies or troops in march- 














CAD AS TRA L S UR VE VS. 


103 


ing, and for resting and camping places for convoys and supply 
trains. In the memoir various streams must be noted, their 
number, position, depth, banks, fords, bridges etc. ; ponds, 
marshes, canals, and springs must all be described, with a state¬ 
ment as to how they are formed, whether subject to overflow, 
and if crossed by roads, how and where. Bodies of woodland 
and forest must be described, as to their shapes, positions, 
etc. The classes of roads, their condition, facilities for 
passage of heavy wagons or troops, and for repair must also 
be noted. 

Villages and fortified places must be described, with notes 
of houses, materials of construction, supply depots, work¬ 
shops, and fortifications. Statistics must also be gathered 
of modes of transportation of horses, wagons, cattle, sheep, 
etc., also of available provisions, including corn and hay 
for forage. Mountains and hills must be described, with 
regard to their continuity, direction, nature of slopes, how 
covered, their area; also whether rocky or smooth, practical 
for occupation by either arm of the service, and if so, where; 
also, the passes across them, their relation to the main chain 
or ridge, etc., etc. 

44. Cadastral Surveys. —This class of surveys takes no 
account of the surface of the earth outside of the limited area 
covered by the immediate route touched in the actual process 
of fixing located points upon it. A cadastral survey is prose¬ 
cuted for the sole purpose of determining political or property 
lines, and merges on the one hand into surveys crudely 
executed with chain and compass (Arts. 99 and 91), and on 
the other hand into field methods of the most refined 
geodetic nature (Art. 201). Cadastral surveys are mentioned 
here because they are frequently made with such thoroughness 
as to result in the production of a topographic map of the 
entire area inclosed within the boundary lines. 

The word cadastral is one which is not familiarly used in 


104 


MILITARY AND CADASTRAL SURVEYS. 


surveying nomenclature and the meaning of which is variously 
and frequently erroneously interpreted. It is probably of 
French origin and was apparently first applied with any defi¬ 
niteness at the statistical conference held in Brussels in 1853, 
as referring to national maps on very large scales, approxi¬ 
mating 1 12500. At the same time the term “tableau d’as- 
semblage ” was applied to large-scale general maps of, say, 
about 1 : 10,000. The word cadastre has been accepted in 
Great Britain as being referred to a map or survey on a 
large scale, because the scale of the map corresponds with a 
cadrcr, being that scale in nature which will permit of 
representing accurately the width of a road and the dimensions 
of a building. More recently on the Continent the expres¬ 
sion “ cadastral survey ” is applied to a plan from which the 
area of land may be computed and from which its revenue 
may be valued. 

As now more generally understood, a cadastral survey is 
one which includes several of the above features. It is not a 
topographic survey for representation of a terrane on a very 
large scale, nor is it any form of a topographic survey, as 
defined by the English interpretation of the meaning of 
the word cadrer. It is essentially a property survey as ex¬ 
pressed in the more recent Continental definition, but it is ex¬ 
ecuted not only that the areas of lands may be computed for 
the valuation of revenue, but also and primarily for the pur¬ 
pose of fixing public and private property lines by marks and 
monuments. A secondary result of a cadastral survey of 
such thoroughness as to closely cover the entire area is the 
procurement of such notes as will permit of the making of a 
large-scale topographic map, such as would come under the 
British or Brussels definition. Examples of topographic 
maps resulting from or executed in the progress of cadastral 
surveys are instanced in Article 27, describing the surveys of 
the cities of Baltimore and St. Louis. In both of these cases 


CADASTRAL SURVEYS. 


105 


the cadastral surveys are based on control of geodetic ac¬ 
curacy. 

Near the other extreme are the cadastral surveys executed 
under the direction of the U. S. Land Office in the subdi¬ 
vision of the public lands of the West. The primary object 
of these surveys is the division of the public lands into areas 
called townships and sections by means of comparatively 
crude transit and chain traverses (Art. 87). The result is de¬ 
picted in maps or plots showing the property outlines with 
their dimensions. A secondary result is the furnishing of 
data for the computation of the areas of the various subdi¬ 
visions. A tertiary result is the representation of the terrane 
covered by a crude topographic map of such quality as to be 
exploratory only in its accuracy. The only information ac¬ 
curately depicted on such maps is that lying immediately 
under the route covered by the boundary survey, the infor¬ 
mation contained between such traverse lines being largely 
interpolated by estimation and guess. 

Another step toward the attainment of a higher grade of 
cadastral survey in the subdivision of the public lands is in¬ 
stanced by the mode of subdivision employed in the public- 
land surveys of the Indian Territory as executed by the U. S. 
Geological Survey (Fig. 33). In addition to being performed 
in a manner similar to that of other public-land surveys, these 
surveys are controlled and checked by means of a geodetic 
survey executed by trigonometric methods, thus giving it a 
far more permanent character and fixing with accuracy its 
position upon the face of the earth. Moreover, the resulting 
map is a true topographic or geographic map, because the 
terrane between the property lines was surveyed and its eleva¬ 
tions determined as an adjunct to the execution of the cadas¬ 
tral survey. 

A still more accurate cadastral survey prosecuted solely 
for the purpose of marking political boundary lines is that ex- 


106 MILITARY AND CADASTRAL SURVEYS. 

ecuted by the Massachusetts Topographic Commission. The 
purpose of this survey is the demarkation with exactitude 
upon the ground-surface of town and county boundary lines. 



Fig. 33.—Cadastral Map of U. S. Public Land Survey. Indian 

Territory. 

Original scale 2 inches to 1 mile. 

The field-work of this survey is based upon and controlled by 
a trigonometrical survey of geodetic accuracy, and as a result 
the positions of the various monuments and their connect¬ 
ing lines are fixed on the surface of the earth in proper astro¬ 
nomic position. The primary result of this survey is the ac- 









































CADASTRAL SURVEYS. 


10 / 


curate demarkation of the boundaries. A secondary result is 
the procurement of data by which the exact area of each 
township may be computed. The amount of data gathered, 
however, does not permit of the making of a topographic 
map of the area included within the bounds. 

Table III. 

SCALE AND COST OF CADASTRAL SURVEYS. 


Country, 

Scale. 

Great Britain; Ord- 


nance Survey. 

i:2,500 

City of St. Louis, Mo. 


(Cadastral). 

1: 2,400 

City of Baltimore, 


Md. (Cadastral).... 

1:2,400 

Public Land Subdi- 


visions, U. S. G. S. 

1:31,680 


Relief. 

Cost per sq. mi. 

Hachures and 
contours 

$ 294.OO 

Contours 3 ft. 

739.OO 

“ 5 “ 

4070.00 

“ 50“ 

31.00 












CHAPTER VI. 


TOPOGRAPHIC FORMS. 

45, Relations of Geology to Topography. —The claim is 

not infrequently made by geologists that a knowledge of this 
science is essential to the topographer in the prosecution of 
his work. On the other hand, the not infrequent contention 
of the topographer is that no amount of knowledge of the 
sciences will add one jot to his ability to accurately represent 
the topography. Both contentions are correct under various 
circumstances, dependent upon the purpose of the resulting 

map and the accuracy desired. A knoivledge of geology may 

> 

influence the plans but not the operations of a topographic 
survey. It may aid in the execution of the topography, but 
Cannot exercise a control over it. It has been stated by a 
topographer that the sphere of the sciences is to follow after 
topographic surveying and not to precede it; that they 
furnish resulting utilities dependent upon the topography, 
but are not essential factors controlling its method. This 
latter view is no more correct than the former. 

Where topographic surveying is to be executed with only 
pne or two specific objects in view, as the furnishing of a base 
map for a geologic survey, or of a small-scale topographic 
map for general uses as a guide or sketch map, a knowledge of 
the science of geology may be of the greatest utility. Under 
such circumstances the greatest accuracy is not essential, and 
such accuracy as is required the topographer will attain. His 
work may, however, be greatly facilitated, and the expression 


ORIGIN AND DEVELOPMENT. 


109 


of the resulting map improved, if he has sufficient knowledge 
of both geography and of geology to appreciate the origin 
of the topographic forms which he is sketching, and the 
way in which the various rocks have been upheaved, eroded, 
folded, or deposited. By a careful study of such forms as 
he first encounters the topographer is able to bring out with 
less effort the same characteristics in such similar formations 
as he may encounter as his work progresses. 

On the other hand, if the survey is to result in an accurate 
topographic map useful as a base-map for all scientific and en¬ 
gineering purposes, the topographer must obtain such amount 
of control, and must see at such close range every feature 
which he sketches, that any amount of knowledge of geology 
or geography will add little to the quality of his representa¬ 
tion of the terrane or the accuracy with which the result is 
depicted. If his work is done with that detail which is es¬ 
sential to the making of an accurate map, he will locate con¬ 
tours with such frequency that the resulting map will depict 
them as they actually exist, regardless of theories as to 
the origin of the forms sketched. Such a map is a topo¬ 
graphic map per sc. It is the mother map, and from a study 
of it the geologist or geographer learns to interpret the origin 
of topographic forms and is enabled to devise a correct scien¬ 
tific hypothesis. 

46. Origin and Development of Topographic Forms.— 

A knowledge of the laws governing the origin and develop¬ 
ment of topographic forms is desirable in those who would 
intelligently depict them. The new topographic method 
demands such a knowledge in order that the surveyor may 
attain the highest skill, not only in the representation of the 
relief, but in the speed and methods, and consequently the cost, 
of such representation. The various rules forthe classification 
of topographic forms hold good only in limited areas, and 
are subject to so many exceptions that any attempt at their 
general application utterly fails. Nevertheless a knowledge 


IIO 


TOPOGRAPHIC FORMS. 


of these may frequently aid in mapping a region to which 
they are found to apply. 

Prof. John C. Branner has aptly described topography as 
the “ expression of geologic structure, much as the outlines of 
the human body express anatomical structure.” As topo¬ 
graphic form is the resultant of eroding agencies and the re¬ 
sistance of rocks, their study, he says, belongs fundamentally 
to the province of geology. It follows that for a thorough 
understanding of topographic forms the surveyor should have 
a knowledge of geology. This is true in topographic survey¬ 
ing, because a large part of every map must be sketched in 
(Arts. 9 and 13), and this sketching cannot be properly done 
unless the surveyor possesses some knowledge of the forma¬ 
tions which he is depicting. Unless he knows what to look for 
he does not find it all, but only a part of it; consequently it is 
of importance to the topographer that he should know what 
kind of topography to expect, and to this end the more he 
knows of the materials in which the topography is carved, and 
the agencies which shaped it, the clearer will be his insight and 
the less the waste of energy and time required for the repre¬ 
sentation of the relief. 

47. Physiographic Processes. —Before an intelligent un¬ 
derstanding can be had of the origin of topographic forms we 
must first look to the processes in nature by which sudi 
forms are created. Major J. W. Powell defines Physiography 
as a description of the surface features of the earth, and a 
study of Physiography as including an explanation of their 
origin. 

The earth has three moving, envelopes: 

1. The atmosphere which covers it to a great depth; 

2. Water, which covers more than three-fourths of its 
surface; and 

3. A garment of rock 

There are two general classes of topographic agencies , 
which may be called constructive and destructive . An ex- 


PH YSIOGRA PH/C PR0CESSES. 


I I I 


ample of the former is a volcanic cone built of ejecta from 
the vent of a volcano which may have burst forth upon a level 
plain, or of a plain resulting from the flow of fluid lavas, 
which form a flat surface by filling up existing irregularities. 
The destructive agencies, chief among which are erosion by 
running water, wave action, wind, and frost, would be illus¬ 
trated by a part of the ocean’s bottom, which being uncovered 
and left as dry land would possess certain irregularities, but of 
a smoothed-out and easily rolling character. Erosion and 
wave action soon begin to attack such a surface and to cut 
stream-beds and produce topographic forms altogether differ¬ 
ent from its original surface. 

Among the chief constructive agencies are: 

1. Disastrophic processes by which regions sink and rise; 

2. Vulcanic processes, due to ejectment from rents in the 
earth’s surface of material brought from the interior. 

Among the chief of the destructive agencies are : 

1. Aqueous erosion, due to water flowing over the surface 
of the earth, as from rain, springs, or streams; 

2. Aerial erosion, from wind-driven sand; 

3. Corrasion, due to ripple and wave action and to gla¬ 
ciers ; and 

4. Disintegration, due to changes in temperature and to 
frost. 

Horizontal changes are produced primarily by aqueous 
agencies, and the action of water is the chief agency in 
shaping topographic forms. Aqueous agencies act by erosion, 
transportation, and corrasion, and of these erosion has pro¬ 
duced nine-tenths of the topographic forms in the United 
States. To the topographer the forms produced by aqueous 
erosion are those commonly seen, and have been classed by 
Mr. Henry Gannett as regular forms, while those shaped by 
other agencies he calls irregular. Aqueous erosion, being 
produced by simple actions of a kind which can be seen and 
comprehended, produces forms which can be to a certain ex- 


I 12 


TOPOGRAPHIC FORMS. 


tent predicted or foreseen. The forms produced by other 
agencies, being unseen, can rarely be predicted. Such agen¬ 
cies have produced the complicated system of mountain-folds 
of the Appalachian region. Acting on these complicated 
forms, aqueous erosion has in the same regions produced a re¬ 
markably complex drainage system. 

48. Classification of Physiographic Processes. —Physio¬ 
graphic processes, by which is meant the operations of nature 
by which topographic forms are produced, may be divided 
into four classes: 

1. Diastrophism, 

2. Vulcanism, 

3. Weathering, 

4. Gradation. 

These various primary physiographic processes have been 
well classified by Doctor C. Willard Hayes in the following 
tabular form : 

I. Diastrophism. 

1. Tangential forces, producing deformations of the strata. 

2. Radial forces, producing vertical oscillations. 

II. Vulcanism. 

1. Intrusions. 2. Eruptions. 


(1) Plutonic plugs. 

(2) Laccolites. 

(3) Volcanic necks. 

(4) Dikes. 


(1) Lava flows. 

(2) Explosive ejections. 


III. Weathering. 

1. Agencies, Chemical. 


Agencies, Mechanical. 

(1) Heat. 

(2) Moisture. 

(3) Vegetation. 


(1) Hydration. 

(2) Oxydation. 

(3) Solution. 


2 . Conditions. 


3. Effects upon. [amorphic rocks. 

(1) Igneous and crystalline met- 

(2) Calcareous rocks. 

(3) Argillaceous rocks. 

(4) Siliceous rocks. 


(1) Altitude. 

(2) Temperature. 

(3) Humidity. 


EROSION, TRANSPORTATION, A ND CORE A SION. 11 3 


IV. Gradation. 

1. Agencies. 

(1) Running water. (2) Glaciers. 

a. Erosion. (3) Winds. 

b. Corrasion. 

c. Planation. 


2 . 




3 - 


4 - 


Processes. 

(1) Disintegration. 

a. Chemical. 

b. Mechanical. 


(2) Transportation. 

a. By solution. 

b. By suspension. 

c. By rolling. 


(3) Deposition. 

a. Alluviation. 

b. Sedimentation. 

c. Chemical precipitation. 
Conditions modifying Gradation. 

(1) Rainfall. (2) Declivity—effect on. 

a. Amount. a. Corrasion. 

b. Distribution. b. Transportation. 

( 3 ) Vegetation—effect on. 

a. Erosion. 

b. Rainfall. 

Drainage development. 

(1) The normal cycle—characteristics of. 

a. Youth. b. Maturity. c. Old age. 

(2) Consequent streams—courses determined; 

a. By accidental irregularities. 

b. By deformation before emergence. 

c. By deformation after emergence. 

(3) Antecedent streams. 

(4) Superimposed streams; 

a. From unconformable horizontal strata. 

b. From planation and alluvial deposits. 

(5) Subsequent streams. 

A. Conditions favoring stream adjustments. 

a. Successive periods of base leveling. 

b. Strata of diverse resistance. 

c. Folded structure. 

d. Local deformations of base level. 

B. Process of stream diversion. 


49. Erosion, Transportation, and Corrasion.—The ero¬ 
sive action of water on rocks has been enormous in amount 
and has continued through such extended periods of time a 


TOPOGRAPHIC FORMS . 


114 

to carve the giant ranges of Colorado from enormous plateaus. 
From these plateaus the drainage systems of the Colorado 
and Arkansas rivers have been worn away to such depths as 
to produce canyons and cliffs of thousands of feet in depth 
(Fig. 34)0 The action of water through weathering is illustrated 
in the disintegration of rock and its conversion into soil. 
Transportation has carried the material thus loosened. In the 
movement of this transported material by streams it has cor- 
raded other materials from their channels. It is thus seen that 
corrasion is effected by the detritus which running water holds 
in suspension. The rate of corrasion is increased in propor-' 
tion to the volume of the stream, its velocity, and the amount 
of detritus borne, as well as by the coarseness of that detritus. 

If a stream have its source at a high altitude and be as¬ 
sumed to have a uniform slope thence to its mouth, its vol¬ 
ume, velocity, and amount of detritus borne will be greatest 
near its mouth, and there corrasion will be most rapid. Asa 
result the slope of the stream will be reduced rapidly near its 
mouth, producing as a normal profile of its bed a curve con¬ 
cave upward. While the slope of the bed remains great and 
the velocity consequently great, the stream has a compara¬ 
tively straight channel. As the slope is reduced the course 
of the stream becomes crooked and winding and its corrasive 
agencies are diverted from its bottom to its sides. Therefore, 
swift streams flow in straight channels , sluggish streams in 
crooked channels. This operation is being performed not 
only in the main stream, but in its numerous affluents to the 
minutest rill, but with different intensity. Accordingly, the 
higher branches have less power to cut and level the surface, 
and there the curves of their bed are convex upward. Such a 
curve is the curve of the terrane, while the concave curve is 
that of the watercourse. The former is the curve of the 
upper relief of high slopes, and the latter of the valleys. 

In an arid regioii rainfall is scanty and spasmodic, stream- 
beds are few in number, and the drainage system is conse¬ 
quently imperfectly developed. There the erosion of the 


Fig. 34.—Canyon in Homogeneous Rock ; also Valley, Cliff, Dome, Mountain, River, Creek, Pond, and Fall 

Yosemite Park, Cal. 

Scale 2 miles to 1 inch. Contour interval 100 ft. 










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H 

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JO 

CO O 
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P *t 

<r m 

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z 

3 . ° 

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53“ CO 
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50 

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3-2 

3 § 

< Z 
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w 2 
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EROSION, TRANSPORTATION,. AND CO RR A SION. I 19 

terrane is slow, while stream corrasion-is proportionately rapid 
because such rainfall as occurs is in sudden showers of great 
volume but short duration. It is thus that the canyons of 
the arid regions have been formed. 

There is a tendency in every stream to extend its drainage 
area by erosion on all sides, the stream having the most 
rapid fall eroding its margin most quickly. Hence the stream 
having the most rapid descent draws area from others. This 
extension of drainage basins is called piracy and is in active 
progress in the Appalachian Mountains (Fig. 35). While 
under some circumstances the courses of streams are change¬ 
able, under others they maintain their courses with great 
persistency. An example of the latter condition is seen in water- 
gaps (Fig. 35) and canyons. A canyon illustrates the persistency 
of stream channels (Fig. 34); it is the result of the uplift of 
a mountain range across the course of an existing stream. The 
rate of uplift has been such that the stream has been able to 
maintain its course by corrasion as the mountain rose. Wi?id- 
gaps illustrate the changeability of stream channels, since they 
are abandoned water-gaps from which the stream has been 
drained by a more powerful pirating neighbor. These are to be 
clearly distinguished from passes in mountain ranges caused 
by the erosion of divides at stream-heads. 

Since disintegration of hard or of insoluble rocks goes on 
slowly, and of soft or soluble rocks rapidly, elevated areas 
due to erosive action are as a rule composed of the former, 
and depressed areas resulting from the same kinds of action 
are generally composed of the latter class of rocks. Streams 
usually make their channels along lines of least resistance. 
The level' surface of the plateau is generally the summit of 
the hard stratum of rock from which, perhaps, softer strata 
have been eroded. Other things being equal, the harder the 
rock the steeper the slope , the softer the rock the more 
gentle the slope. Applying this principle to the cross- 
section of two stream-beds, one in soft rock, the other in 
hard rock, they will be carved into forms shown in Figs. 


120 


TOPOGRAPHIC FORMS. 


36 and 37, in which the lines indicate progressive stages. 
Where the rock strata are laid in horizontal beds, alter¬ 
nately soft and hard, the forms resulting from stream corra- 
sion will be similar to those represented in Figs. 38 and 39. 
Where one or more horizontal beds are of nearly equal hard- 



Fig. 36.—Erosion in Soft Rock. 



Fig. 37.—Erosion in Hard Rock. 



Fig. 38.—Erosion in Horizontal Beds ok Hard and Soft Rock. 



Fig. 39.—Erosion in Alternate Beds of Soft and Hard Rock. 


h 

Fig. 40.—Erosion in Soft Rock Underlain by Hard. 

ness, as .sandstone underlain by granite, the canyon resulting 
from erosion will be similar in form to that shown in Fig. 40. 
If the stream flow parallel to the strike of the inclined bed, 
similar forms will be produced, but on sloping surfaces. 

50. Topographic Forms. —As there are two general 
classes of physiographic processes, the constructive and the 
destructive, so are there two classes of topographic forms 
resulting from these kinds of agencies. 
































































































































































































































































































































































































































TOPOGRAPHIC FORMS. 


I 2 I 


Among the important constructive features are: 

1. Subaqueous forms, resulting from depositions of what¬ 
ever kind in lakes and oceans, as spits and bars erected by 
wave action; 

2. Emergent forms , or those partly built up beneath the 
water but gradually rising above it, as deltas and storm-spits; 

3. Subaerial forms, or those produced by volcanic eject¬ 
ment of dust and ashes which may be blown about, and 
the outpouring of geysers and deposits from certain saline 
springs; and 

4. Subsurface forms, as faults and folds, produced by 
slipping and distortion, and crumbling of the earth’s crust. 

Among the chief destructive features are : 

1. Vertical diastrophic forms , resulting from upheaval 
and subsidence, as massive mountain-ranges, plateaus, and 
emerged plains. 

2. Horizontal diastrophic forms, resulting from deforma¬ 
tions of the strata due to tangential forces, as faults, flexures, 
throws, and folds. 

\ 

3. Vulcanic intrusions , resulting from materials brought 
to the surface from the interior, as dikes, laccolites, and 
volcanic necks. 

4. Vulcanic ejecta, due to materials of the interior vio¬ 
lently erupted by volcanoes or geysers, as lava-plains, cinder- 
cones, deposits from spring-waters. 

Topo'- aphic forms are modeled— 

1. By the character and hardness of the rocks; 

2. By the geologic structure or position of bed-rock; 

3. By the slope of the surface; 

4. By climatic conditions; 

5. By accidents during development; 

6. Bv length of time during which eroding agents have 
acted ; and 

7. By the nature and working methods of these agencies. 

Topographic forms vary according to the physiographic 


122 


TOPOGRAPHIC FORMS. 


processes which have produced them. According as these 
act they may be divided into two great classes: 

1. By vertical change; 

2. By horizontal change. 

As a result of the former the surface of the earth moves 


up and down, producing the general forms which are said to 
result from uplift and downthrow. As a result of horizontal 
change land is transported from one locality to another. 
This form of change acts generally through the agency of 
water and, to a minor extent, of the wind. It produces the 
general forms resulting from aggradation and degradation, 
and through the action of erosion, corrasion, and trans¬ 
portation. 

51. Classification of Topographic Forms.—Doctor 

Hayes has classified topographic forms in the following 
simple tabular manner: 


I. Co 7 istructionalforms due to 

1. Diastrophism. (Fig. 21.) 

(1) Emerged plains. 

(2) Plateaus. 

(3) Block ranges. 

(4) Lake basins. 

II. Gradationalfor)ns due to 
I. Aggradation by 

(1) Water. (Fig. 42.) 

a. Alluvial cones. 

b. Alluvial plains. 

c. Deltas. 


( 3 ) 


2. Vulcanism. (Fig. 41.) 

(1) Lava plains and coulees. 

(2) Volcanic cones. 


(2) Ice. (Fig. 41.) [drift-sheets. 

a. Moraines— (a) terminal, ( b ) 

b. Eskers. 

c. Sand plains. 

d. Drumlins. 


Wind. (Fig. 43.) 

a. Dunes. 

2. Degradation. 

(1) Sculptured forms. 

(Figs. 4, 34 and 44.) 

a. Canyons and gorges 

b. Valleys. 


b. Loess plains. 

(2) Residual forms. 
(Figs. 6. 44 and 45.) 

a. Plateaus. 

b. Mesas. 


c. Plains and peneplains, c. Mountains. 1 

d. Lake basins. d. Ridges, hogbacks. 

e. Hills. 



Fig. 41._Volcanic Mountain, Crater, Cinder-cone, Glacier, and 

Moraines, Mt. Shasta, Cal. 

Scale 4 miles to i inch. Contour interval 200 ft. 

1 2 3 






























. 











. 






. 
























































































































































Fig. 42.—Alluvial 
Scale 1 


Ridge, Crevasse, Swamp, River, and Flood Plain 
Lower Mississippi River. 
mile to I inch. Contour interval 5 ft. 



1 2 5 





































































































































































































































* 


V 























Glasgow 




North Pt 


Norik l >en< 


Yim-uw 


Btxutei 


Cra wford 


*G. 5 '. 5:0 


Marsh 


Fig. 4.3.—Sand-dune, Spit, Ocean, Bat, Lagoon, Slough, Tidal Flat, 
Swamp, and Marsh, Coos Bay, Ore. 

Scale 2 miles to 1 inch. Contour interval 100 ft. 


1 27 




















































































. 





















































































Fig. 45. —Mountain Range, Amphitheatre, Scarp, Basin, Peak, and Creek, Irwin, Colo 

Scale 1 mile to 1 inch. Contour interval 100 ft. 













































. 

. 







' 








. 












1 





















• 






























































GLOSSARY OF TOPOGRAPHIC FORMS. 


The following list of definitions is intended to include all 
those terms employed popularly or technically in the United 
States to designate the component parts of the surface of the 
earth. None of the words similarly applied in other portions 
of the world are given. So far as practicable, the endeavor 
has been to refrain from defining such words or using such 
definitions as refer merely to the origin of the various topographic 
forms. At the same time it has been found necessary in a few 
instances to define forms according to their variety or origin, 
as those resulting from volcanic or glacial action. In the case 
of names which are locally peculiar to a limited portion of the 
country, the effort has been to indicate the regions in which they 
were employed. The language whence derived is denoted by 
Sp. for Spanish, Fr. for French, etc.; the word “origin” follow¬ 
ing indicates that it has been generally adopted in American 
nomenclature. 

Acclivity: An ascending slope as opposed to declivity. 

Aiguille: A sharp rocky peak or needle. (Fr. origin.) 

Alkali Flat: A playa; the bed of a dried-up saline lake, the soil of which 
is heavily impregnated with alkaline salts. 

Alpine: Pertaining to mountains of great height and ruggedness of outline 
and surface, and containing glaciers. Resembling a great mountain 
range of southern Europe called the Alps. 

Amphitheater: A cove or angle of glacial origin near the summit of a high 
mountain and nearly surrounded by the highest summits. A small 
flat valley or gulch-like depression at the head of an alpine mountain 
drainage. Local in far West. 

Anse : A cove. (Fr.) 


'33 


*34 


GLOSSARY OF TOPOGRAPHIC FORMS. 


Arete: A sharp, rocky crest; a comb-like secondary crest of rock which 
projects at a sharp angle from the side of a mountain. (Fr. origin.) 

Arm : A branch or bay of a lake or sea. 

Arroyo: The channel of an intermittent stream steep cut in loose earth; a 
coulee. Local in Southwest. (Sp.) 

Artesian Well: A well which has been excavated or drilled through imper¬ 
vious strata to a subterranean water supply which has its source at a 
higher level. The resulting hydrostatic pressure causes the water to 
rise in the bore to a sufficient height to overflow at the mouth of the 
well. 

Atoll: A ring-shaped coral island nearly or quite encircling a lagoon. 

Badlands: Waste or desert land deeply eroded into fantastic forms. Local 
in arid Northwest. 

Backbone : The highest portion or elongated crest of a ridge. 

Backwoods : The wild country back from settlements. 

Bald: A high rounded knob or mountain top, bare of forest. Local in 
Southern States. 

Bank: A low bluff margin of a small body of water. 

A mound-like mass of earth. 

Bar: An elevated mass of sand, gravel or alluvium deposited on the bed 
of a stream, sea or lake, or at the mouth of a stream. 

Barranca: A rock-walled and impassable canyon. Local in Southwest. 
(Sp.) 

Barren : Open plain and bogs covered with low brush. 

Barrier Beach: A beach separated from the mainland by a lagoon or marsh. 

Barrier Island: A detached portion of a barrier beach between two inlets. 

Base-level Plain: A flat, comparatively featureless surface or lowland result¬ 
ing from the nearly completed erosion of any geographic area. 

Basin: An amphitheater or cirque. Local in Rocky Mountains. 

An extensive, depressed area into which the adjacent land drains, 
and having no surface outlet. Use confined almost wholly to the arid 
West. 

The drainage or catchment area of a stream or lake. 

Bay: An indentation in the coast line of a sea or lake; a gulf. 

Baygall: A swamp covered with growth of bushes. Local on South Atlantic 
Coast. 

Bayou: A lake or intermittent stream formed in an abandoned channel 
of a river; one of the half-closed channels of a river delta. Local on 
Gulf Coast. (Fr. origin.) 

Beach: The gently sloping shore of a body of water; a sandy or pebbly 
margin of water washed by waves or tides. 

Bed: The floor or bottom on which any body of water rests. 

Bench: A strip of plain along a valley slope. 

A small terrace or comparatively level platform on any declivity. 




GLOSSARY OF TOPOGRAPHIC FORMS. 


T 35 


Bend : A sharp turn in a stream. 

Bight: A small bay. 

Bluff: A bold, steep headland or promontory. 

A high, steep bank or low cliff 

Boca: A mouth; the point at which a stream way or drainage channel 
emerges from a barranca, canyon or other gorge, and debouches on a 
plain. (Sp.) 

Bog: A small open marsh. 

Bogan : A marshy cove by a stream. 

Bolson: A basin; a depression or valley having no outlet. Local in South¬ 
west. (Sp., meaning “purse.”) 

Bottom: The bed of a body of still or running water. 

Bottom Land: The lowest land in a stream bed or lake basin; a flood plain. 

Boulder: A rounded rock of considerable size, separated from the mass 
in which it originally occurred. 

Box Canyon: A canyon having practically vertical rock walls. 

Branch: A creek or brook, as used locally in Southern States. Also used 
to designate one of the bifurcations of a stream, as a fork. 

Breaks: An area in rolling land eroded by small ravines and gullies. Local 
in Northwest. 

Bridal-veil Fall: A cataract of great height and such small volume that 
the falling water is dissipated in spray before reaching the lower stream- 
bed. 

Brook: A stream of less length and volume than a creek, as used locally 
in the Northeast. 

Brow: The edge of the top of a hill or mountain; the point at which a gentle 
slope changes to an abrupt one; the top of a bluff or cliff. 

Butte: A lone hill which rises with precipitous cliffs or steep slopes above 
the surrounding surface; a small isolated mesa. Local throughout far 
West. (Fr.) 

Cajon: A box canyon. Local in Southwest. (Sp., meaning “box.”) 

Cala: A creek. Local in Southwest. (Sp.) 

Caldera: A deep ravine, hollow or depression in a volcanic formation. (Sp. 
origin.) 

Camas: A small upland prairie; a glade; a small park; a small, gently 
sloping prairie, partly wooded and surrounded by high mountain slopes. 
Local in Pacific Northwest. (Sp., meaning “bed.”) 

Canal : A sluggish coastal stream. Local on Atlantic Coast. 

Candelas : A group of candle-like rocky pinnacles. Local in Southwest. 
(Sp.) 

Canyada: A very small canyon Local in Southwest. (Sp.) 

Canyon: A gorge or ravine of considerable dimensions; a channel cut by 


136 


GLOSS A RY OF TOPOGRAPHIC FORMS. 


running water in the surface of the earth, the sides of which are com¬ 
posed of cliffs or series of cliffs rising from its bed. Local throughout 
the far West. (Sp. origin.) 

Cape: A point of land extending into a body of Water; a salient of a coast. 

Carse: A low fertile river bottom. (Scot, origih.) 

Cascade: A short, rocky declivity in a stream-bed over which water flows 
with greater rapidity and higher fall than over a rapid; a shortened 
rapid, the result of the shortening being to accentuate the amount of 
fall. 

Cataract: A waterfall, usually of great volume; a cascade in which the 
vertical fall has been concentrated in one sheer drop or overflow. 

Catchment Basin: The area on which the precipitation of a stream or 
river system falls. The watershed or drainage area of a stream. 

Cave: A hollow space or cavity under the surface of the earth. 

A depression in the ground, by abbreviation from a “cave-in,” as 
used colloquially. 

Cavern: A large, natural, underground cave or series of caves. 

Cay: A key; a comparatively small and low coastal island of sand or coral. 
Local in Gulf of Mexico. (Sp. origin.) 

Ceja: The cliff of a mesa edge; an escarpment. Local in Southwest. (Sp.) 

Cerrito, or Cerrillo: A small hill. Local in southwest. (Sp.) 

Cerro: A single eminence intermediate between hill and mountain. Local 
in Southwest. (Sp.) 

Channel: A large strait, as the British Channel. 

The deepest portion of a small stream, bay or strait through which 
the main volume or current of water flows. 

Chasm: A canyon having precipitous rock walls; a box-canyon. 

Cienega: An elevated or hillside marsh containing springs. Local in South¬ 
west. (Sp.) 

Cirque: A glacial amphitheater or basin. (Fr. origin.) 

Clearing : A place whence woods have been removed for a farm. 

Cleugh: A ravine with precipitous rocky sides, containing a torrential stream. 
(Scot, origin.) 

Cliff: A high and very steep declivity. 

Clove: A gorge or ravine. Local in Middle States. (Dutch origin.) 

Coast: The land or the shore next to the sea. 

Coastal Marsh: A marsh which borders a seacoast and is usually formed 
under the protection of a barrier beach. 

Coastal Plain: Any plain which has its margin on the shore of a large body 
of water. 

Col: A short ridge connecting two higher elevations; a narrow pass joining 
two valleys. 

Colina: A hillock or small eminence. 



i>00 


Plate III. —Sand Hills, Bench, Terrace, Creek, and River, above 

Albany, N. Y. 

Scale I mile to l inch. Contour interval 20 ft. 



























































































































































































GLOSS A R Y OF TOPOGRAPHIC FORMS. 


13 7 


Collado: A hill or small eminence (Sp. origin.) 

Colline: A colina. (Obs.) 

Cone: A low, conical hill, built up from.the fragmental material ejected 
from a volcano. 

Continental Shelf: A comparatively shallow marginal ocean bed or floor 
bordering a continent; a submerged terrace bordering a continent 

Coombe: A hollow in a hillside above the most elevated spring. 

Cordillera: A group of mountain ranges, including valleys, plains, rivers, 
lakes, etc.; its component ranges may have various trends, but t. c 
cordillera will have one general direction. (Sp. origin.) 

Corner : A settlement at a cross-roads. 

Corrie: A circular hollow on a hillside whence flows a spring. See Coombe. 
(Scot, origin.) 

Coteau: An elevated, pitted plain of rough surface. Local in Missouri 
and neighboring Sta es. (Fr. origin.) 

Coulee: A cooled and hardened stream of ’ava. Coulees occur as ridges 
of greater or less length and dimensions, but rarely of great height. 
Local in Northwest. 

A wash or arroyo through which water flows intermittently. Local 
in Northwest. (Fr. origin.) 

Cove: A small bay. 

An amphitheater or indentation in a cliff. It may be the abrupt 
heading of a valley in a mountain. 

Crag: A rough, steep or broken rock standing out or rising into prominence 
from the surface of an eminence; a rocky projection on a cliff or ledge. 

Crater: The cup-shaped depression marking the position of a volcanic vent; 
its margin is usually the summit of the volcano. 

Creek: A stream of less volume than a river. 

A small tidal channel through a coastal marsh. 

Crest: The summit land of any eminence; the highest natural projection 
which crowns a hill or mountain. 

Crevasse: A fissure n a glacier. 

A break in a levee or other stream embankment. (Fr. origin.) 

Cuesta: An ascending slope; a tilted plain or mesa top. Local in Southwest. 
(Sp.) 

Current: A continuous movement or flow, in one direction, of a body of 
water; a stream in or portion of an ocean which has continuous motion 
or flow in one direction. 

Cusp : A point of land extending into the water and bounded by two 
concave beaches. 

* 

Dale : A vale or small valley. 

Dalle: A rapid. Local in Northwest. (Fr. origin.) 

Deadwater: Part of a stream having very sluggish or no current. A 
Stillwater. 


I3» 


GLOSSARY OF TOPOGRAPHIC FORMS. 


Declivity: A descending slope, as opposed to acclivity. 

Deep: A profound or abysmal depression in the ocean bottom. 

Defile: A deep and narrow mountain pass. 

Delta: The low alluvial land about the mouth of a river which is divided 
down-stream into several distributaries. 

Depression: A low place of any dimensions on a plain surface; the neg¬ 
ative or correlative of elevation or relief. 

Desert: An ari.d region of any dimensions, barren of water other than in 
occasional flood streams or springs, frequently covered with consid¬ 
erable growths of cacti, coarse bunch grass, mesquite and other shrubs. 
A desert is not necessarily a plain surface, as most deserts are broken 
by the sharp escarpments and buttes which are common to the arid 
regions, by sand dunes or volcanic ejecta. A desert may include 
canyons and mountains of considerable difference,of elevation. 

Devil’s Slide, Head, etc.: Applied to topographic features of unusual, 
uncanny or forbidding aspect. 

Dike: A ridge having for its core a vertical wall of igneous rock. 

Divide: The line of separation between drainage systems; the summit of 
an interfluve. 

The highest summit of a pass or gap. 

Doab: A low-lying tongue of land between two confluent streams. (East 
Indian.) 

Dome: A smoothly rounded rock-capped mountain summit. 

Down: A sandy, undulating, or hilly tract near the seashore, covered with 
fine turf. (Br. origin.) 

Draft: A draw. 

Drain : A small channel or depression, which carries off the run-off from a 
slope. 

Drainage Area: The watershed of a given stream or river system. The 
catchment basin of a stream. 

Draw: A very shallow and small gorge, gulch or ravine; the eroded channel 
through which a small stream flows. 

Drift: A slow, great ocean current. 

Drumlin: A smooth, oval or elongated hill or ridge composed chiefly of 
glacial detritus. 

Dry Wash: A wash, arroyo or coulee in the bed of which is no water. 

Dune: A hill or ridge of sand formed by the winds near a sea or lake sho re, 
along a river-bed or on a sandy plain. 

Eddy: A deep cove in a stream below a bar, where water circles around. 

Eminence: A mass of high land. 

Escarpment: An extended line of cliffs or bluffs. 

Eschar: An esker. 

Esker: A long, winding ridge of sand or gravel, the deposit from a stream 
flowing beneath a glacier. 


GLOSSARY OF TOPOGRAPHIC FORMS. 


139 


Estuary: A river-like inlet or arm of the sea. 

Everglade: A tract of swampy land covered mostly with tall grass. Local 
in South. 

Fall: A waterfall or cataract. 

The flow or descent of one body of water into another. 

Fan: A mountain delta; a conical talus of detrital material. 

Fen: A boggy, marshy land. (Br. origin.) 

Fiord: A narrow inlet with high, rocky walls; a glacial gorge filled by an 
arm of the sea. 

Flat: A small plain usually situated in the bottom of a stream gorge; often 
applied to a small area of tillable land in the bend of a bluff-walled 
stream. 

Floodplain: Any plain which borders a stream and is covered by its waters 
in time of flood. 

Floor: The bed or bottom of the ocean. 

A comparatively level valley bottom. 

Fly: Corrupted from vly. 

Foot: The bottom of a slope, grade or declivity. 

Foothill: One of the lower subsidiary hills at the base of a mountain. 

Foreland: A promontory or cape. 

Fork: One of the major bifurcations of a stream; a branch. 

Forks : The place of branching of a stream, looking up stream. 

Fountain: A flow of water rising in a jet above the surrounding surface. 
Artesian wells, geysers and springs may be fountains. 

Frith: A rock girt, narrow arm of the sea, a fiord. (Scot, origin.) 

Fumarole: A spring or geyser which emits steam or gaseous vapor; found 
only in volcanic areas. 

Gap: Any deep notch, ravine or opening between hills, or in a ridge or moun¬ 
tain chain. 

Gate . A narrow opening between rocks or headlands, admitting from one 
valley or one bay to another. 

Geyser: A hot spring, the water of which is expelled with steam in an accu¬ 
mulated volume in paroxysmal bursts. 

Glacial Gorge: A deeply cut valley of U-shaped cross-section, the result 
of glacial erosion. 

Glacial Lake: A lake, the basin of which has been carved by glacial action, 
or is dammed on one side by glacial detritus. 

Glacier: A permanent body or stream of ice having motion. 

Glade: A grassy opening or natural meadow in the woods; a small park. 
Applied in western Maryland to a brushy, grassy or swampy open¬ 
ing in the woods. 

Glen: A secluded and small narrow valley. 


140 


GLOSSARY OF TOPOGRAPHIC FORMS. 


Gorge: A canyon; a rugged and deep ravine or gulch. 

Grade: A slope of uniform inclination. 

Grotto: A small, picturesque cave. 

Gulch: A small ravine; a small, shallow canyon with smoothly inclined 
slopes. Local in far West and North. 

Gulf: A gorge or deep ravine; a short canyon. Local in Southern States 
and New York. 

A bay, usually of great dimensions. 

Gully: A channel cut by running water; less than a gulch or ravine. 

Gut: A narrow passage or contracted strait connecting two bodies of water. 

Harbor : An enclosed arc of deep water offering safe anchorage for vessels. 

Hammock: A hummock. Local in Southeast. 

Hanging Valley: A high glacial valley, tributary to a more deeply eroded 
glacial gorge or fiord. 

Haugh: A low-lying meadow beside a river. (Scot, origin.) 

Headland: A promontory. 

Heath • Bogs or barrens. (Eng.) 

Height of Land: The highest part of a plain or plateau; or, on a highway, 
a pass or divide. Local in Northeast and British America. 

Highland: A relative term denoting the higher land of a region; it may 
include mountains, valleys and plains. 

Hill: An eminence less than a mountain rising above the surrounding land. 

Hillock: A small hill. 

Hogback: A steep-sided ridge or long hill; used to describe a group of sharply 
eroded low hills. 

A steep foothill having parallel trend to the associated moun¬ 
tain range. Local in the far West. 

Hole: A small pond in a swamp or marsh. 

Holl: A small bay, as Wood’s Holl, Mass. Local in New England. 

Hollow: A small ravine; a low tract of land encompassed by hills or moun¬ 
tains. 

Hook: A low, sandy peninsula terminating in curved or hook-shaped end 
forming a bay. 

Hot Spring: A spring, the water of which has a temperature considerably 
above that of its surroundings. 

Huerfano: A solitary hill or cerro. Local in Southwest. (Sp., meaning 
“orphan.”) 

Hummock: A more or less elevated piece of ground rising out of a swamp, 
often densely wooded. 

Inlet: A small narrow bay or creek; a small body of water leading into 
a larger. 

Intercolline : Between hills; especially valleys between volcanic cones. 

Interfluve: The upland separating two streams having an approximately 
parallel course. 

Island: An area of land entirely surrounded by water. In dimensions 


GLOSSARY OF TOPOGRAPHIC FORMS. 


14 r 

islands range from a point of rock rising above the surface of the 
water to an area of land of continental dimensions, as Australia. An 
area of dry land in a swamp. 

Isthmus: A narrow strip of land connecting two considerable bodies of 
land. 

.. - 1.4. 

Kame: A small hill of gravel and sand made by a glacier. 

Karroo : A dry tableland, often rising terrace-like to higher elevations. (So. 
Africa.) 

Kettle Hole: A steep-sided hole or depression in sand or gravel; a hole 
in the bottom of a stream or pond. 

Key: A cay, as the Florida Keys. 

Kill: A creek. Locally in Middle States. (Dutch origin.) 

Knob: A prominent peak with rounded summit. Local in Southern States. 

Knoll: A low hill. 

Lagoon : A shallow bay cut off from a sea or lake by a barrier; often stagnant, 
with ooze bottom and rank vegetation. It may be of salt or fresh water. 
Locally in South and Southwest. (Sp. origin.) 

Lake: Any considerable body of inland water. 

Lakelet : A very small lake. 

Landing : The place where farm or country roads reach a navigable river 
bank. A place where a boat may tie up at the shore of a stream. 

Landslide : Earth and rock which has been loosened from a hillside by 
moisture or snow, and has slid or fallen down the slope. 

Landslip: A landslide of small dimensions. 

Lateral Moraine: A moraine formed at the side of a glacier; usually ridge-like 
in shape. 

Ledge: A shelf-like projection from a steep declivity; a rocky outcrop or 
reef. 

Lenticular Hill: A short drumline. 

Levee: An artificial bank confining a stream channel. (Fr. origin.) 

Littoral : That portion of a shore washed by or between high and low water. 

Llano : An extensive plain with or without vegetation. (Sp.-American origin.) 


Malpais : A badland. Local in Northwest. (Sp.) 

Mareis: A marsh. (Obs. Br. origin.) 

Maremme: A low marshy tract along the coast. (It. origin.) 

Marsh: A tract of low, wet ground, usually miry and covered with rank 
vegetation. It may at times be sufficiently dry to permit of tillage 
or of having hay cut from it. It may be very small and situated high 
on a mountain, or of great extent and adjacent to the sea. 

Meadow: A bit of natural grassland in wooded mountains; a glade or small 
park. 


142 


GLOSSARY OF TOPOGRAPHIC FORMS . 


Medano: A sand dune on the seashore. 

Mesa: A tableland; a flat-topped mountain bounded on at least one side 
by a steep cliff; a plateau terminating on one or more sides in a steep 
cliff. Local in Southwest. (Sp. origin.) 

Mesita: a small mesa. Local in Southwest. (Sp.) 

Minaret: A high, spire-like pinnacle of rock. 

Mire: A small, muddy marsh or bog. 

Monadnock: An isolated hill or mountain rising above a peneplain, after 
the removal by erosion of its surrounding features. 

Monument: A column or pillar of rock. Locally in Rocky Mountain region. 

Moor: A tract of open uncultivable land, boggy or of poor soil, more or 
less elevated. (Br. origin.) 

Moraine: Any accumulation of loose material deposited by a glacier. 

Morass: A swamp, marsh or bog having rank vegetation and muddy or 
offensive appearance. 

Moulin: A vertical cavity, melted and worn in a glacier, down which water 
passes. (Swiss origin.) 

Mound: A low hill of earth. 

Mountain: An elevation of the surface of the earth greater than a hill and 
rising high above the surrounding country. 

Mountain Chain : A series or group of connected mountains having a well- 
defined trend or direction. 

Mountain Range: A short mountain chain; a mountain much longer than 
broad. 

Mountain System: A cordillera. 

Mouth: The exit or point of discharge of a stream into another stream or 
a lake or sea. 

Muskeg: A bog or marsh. Local in Northwest and British America. 

Narrows: A narrow place in a stream. A gorge or narrow pass between 
hills; or connecting two bodies of water. 

Natural Bridge : Any rock bridge spanning a ravine and left in place by natural 
agencies. 

Neck: The narrow strip of land which connects a peninsula with the main¬ 
land. 

Needle: Prominent and sharp rocky pinnacle or spire. 

Ness: A cape or headland, usually a suffix to a promontory name, as Sheer 
ness. (Br. origin.) 

Nev€ : The consolidated granular snow on a mountain summit in which 
glaciers have their source. 

Notch: A short defile through a hill, ridge or mountain. 

Nubble : A small detached mass of rock near a high shore. 

Nullah: A small intermittently dry ravine; an arroyo; a coulee. (East 
Indian.) 

Nunatak: A rock island in a glacier. 


GLOSS A R V OF TOPOG RA PH1C FORMS. 143 

Oasis: A fertile or green spot in a desert. 

Ocean: The great body of water which occupies two-thirds of the surface 
of the earth. The sea as opposed to the land. 

Oceanic Plateau: An irregularly elevated portion of the ocean bed, of con¬ 
siderable extent and perhaps rising in places above the water surface. 

Osar: A ridge of sand or gravel of sub-glacier origin; an esker. 

Outlet: The lower end of a lake or pond; the point in which a lake or pond 
discharges into the stream which drains it. 

Outlier: A rock mass lying beyond the main body from which it has been 
separated by denudation. 

Oxbow: A bend in a stream which turns the latter back on its course. 

Paha: A long ridge of fine, loamy material deposited from a stream which 
has cut a channel in a melting glacier. Local in Iowa and vicinity. 
(Am. Indian.) 

Palisade: A picturesque, extended rock cliff rising precipitately from the 
margin of a stream or lake and of columnar structure. 

Pampas: Vast plains of South America. 

Paramo: A high bleak plateau with stunted trees. (Andes, South America.) 

Park: A grassy, wide and comparatively level open valley in wooded moun¬ 
tains. Local in Rocky Mountains. 

Pass: A gap or other depression in a mountain range through which a road 
or trail may pass; an opening in a ridge forming a passageway. 

A narrow, connecting channel between two bodies of water. 

Passage : A navigable channel between islands. 

Passes: The narrow arms of a delta restricting navigable channels, as the 
passes of the Mississippi. 

Peak: A pointed mountain summit; a compact mountain mass with single 
conspicuous summit. 

Peneplain: A land surface which has been reduced to a condition of low 
relief by the erosive action of running water. 

Peninsula: A body of land nearly surrounded by water. 

Picacho; A peaked butte. Local in Southwest. (Sp.) 

Pinnacle: Any high tower or spire-shaped pillar of rock, alone or cresting 
a summit. 

Pit: Any abrupt and deep depression in the ground surface. 

Pitch: A steep change in a broken slope. 

Pitted Plain: A plain of gravel or sand with kettle holes. 

Plain : A region of general uniform slope, comparatively level, of considerable 
extent and not broken by marked elevations and depressions; it may 
be an extensive valley floor or a plateau summit. 

Plateau: An elevated plain. Its surface is often deeply cut by stream chan¬ 
nels, but the summits remain at a general level. The same topographic 
form may be called a plain and a plateau, and be both. An elevated 
tract of considerable size and diversified surface. (Fr.) 

Platform, Sub-marine : The broad and shallow sea bottom bordering a coast 
and lying terrace-like above the more profound deeps of the sea. 

Playa: An alkali flat; the dried bottom of a temporary lake, without outlet. 
Local in Southwest. (Sp. origin.) 


144 


GLOSSARY OF TOPOGRAPHIC FORMS. 


A small area of land at the mouth of a stream and on the shore of 
a bay; an alluvial flat coast land as distinguished from a beach. Local 
in Southwest. (Sp.) 

Playa Lake: A shallow, storm-water lake. When dried it forms a playa. 
Local in Southwest. (Sp. origin.) 

Plaza: An open valley floor; the flat bottom of a shallow canyon. (Sp.) 

Pocason: A dismal swamp. Local on South Atlantic Coast. (Indian.) 

Point: A small cape; a sharp projection from the shore of a lake, river 
or sea. 

Polder: A track of low land reclaimed from the sea by levees. (Dutch.) 

Pond: A small, fresh-water lake. 

Pool: A water-hole or small pond. 

Portage : A path around a fall or between bodies of water, over which to 
carry boats and stores. 

Pothole: A basin-shaped or cylindrical cavity in rock formed by a stone 
or gravel gyrated by eddies in a stream. 

Prairie: A treeless and grassy plain. 

Precipice: The brink or edge of a high and very steep cliff. 

Promontory: A high cape with bold termination; a headland. 

Puerto: A pass or defile through an escarpment or sierra. Local in Southwest. 
(Sp., meaning “gate.”) 

Quagmire: Any mire or bog. 

Quebrado: A canyon of rugged aspect; a fissure-like ravine or canyon. 
Local in Southwest. (Sp.) . i 

Quickwater : Swift running water unbroken by rapids, 

Rapid : Any short reach of steep slope between two relatively quiet 
reaches in a stream-bed. The water flows over a rapid with greater 
velocity than in adjacent portions of a stream. 

Ravine : A gulch; a small gorge or canyon, the sides of which have com¬ 
paratively uniform slopes. 

Reach : A long stretch in a river where a vessel can make long tacks. 

Reef: A ridge of slightly submerged rocks. 

A ledge of rock on a mountain. 

Relief: Elevation as opposed to depression; the elevated portions of the 
land surface; the irregularities of the earth’s surface. 

Ridge: The narrow, elongated crest of a hill or mountain; an elongated 
hill. 

Riffle: The shallow water at the head of a rapid; a rapid of comparatively 
little fall, as when over gravel bars. 

Rift: A narrow cleft or fissure in rock. 

Rill: A very small trickling stream of water, less than a brook. 

Rincon: Corner or cove; the angular indentation in a mesa edge or escarp¬ 
ment in which a canyon heads. Local in Southwest. (Sp. origin.) 

Rio: A river. Local in Southwest. (Sp. origin.) 


k 




r j’JfiU T 
J V ihj y 




GLOSSARY OF TOPOGRAPHIC FORMS. 145 

Rips or Ripples : See Riffle. 

River : A large stream of running water. A stream of such size as to be 
called a rivet in one locality may be called a creek or brook in another. 

Rivulet: A small river. 

Roaas : A broad and commodious land-locked harbor. 

Rock : A good sized hill of solid rock. 

Rock Cave : A shelter cave. 

Rolling Land: Any undulating land surface; a succession of low hills 
giving a wave effect to the surface. 

Rm . A brook or small creek. Local in South. A small stream connect¬ 
ing two lakes. 

Runnel: A little brook. 

Salient: An angle or spur projecting from the side of the main body of any 
land feature. 

Sand Dune: Any dune. 

Sandia: An oblong, rounded mountain mass. Local in Southwest. (Sp., 
meaning “watermelon.”) 

Savanna: A grassy plain composed of moist and fertile land. Local in South. 

Scarp: An escarpment. 

Scaur: A steep rocky slope; a bare place on a mountainside. 

Sea: A large body of salt water. 

S a Wa 1: A ridge of boulders or sand thrown up by the sea, often en¬ 
closing a lake 

Seep: A small, trickling stream. Local in Southwest. 

Serrate : The rocky summit of a mountain having a sawtooth profile; a 
small sierra-shaped ridge. Local in Southwest. (Sp.) 

She f: A level and rocky terrace of small extent, set high on a hillside. 

Shelter Cave: A cave only partially underground, which is formed by a 
protecting roof of overlying rock; generally open on one or more sides. 

Shoal: A shallow place in a stream or lake; an elevated portion of the bed 
of a stream, lake or sea which rises nearly to the water surface; a bar. 

Shore: The land adjacent to any body of water. 

Sierra: A rugged mountain range with serrate outline. Local in South¬ 
west and Pacific States. (Sp. origin.) 

Silva: A vast wooded plain. (S. America.) 

Sink : The bottom of an undrained basin. 

Sinkhole : A deep hollow in glacial drift A sink. 

Slash: Swampy land, overgrown with dense underbrush. Local in Northeast. 

Slide: The exposed surface left in the trail of a landslide; the place whence 
a landslide has departed. Local in Northeast. 

Slope: The inclined surface of a hill, mountain, plateau or plain or any 
part of the surface of the earth; the angle which such surfaces make 
with the level. 

Slough: A freshet-filled channel or bayou; a depression in an intermittent 
stream channel filled with stagnant water or mire. 

Sound: A relatively shallow body of water separated from the open sea 
by an island and connected with it at either end so that through it ther : 
is clear tidal flow. 


l 4 S a 


GLOSSARY OF TOPOGRAPHIC FORMS. 


Spit: A low, sandy point or cape projecting into the water; a barrier beach. 

Spring: A stream of water issuing from the earth. 

Spur: A sharp projection from the side of a hill or mountain; a radial ridge 
of subordinate dimensions. 

Steppe: A vast elevated plain, generally treeless. (Rus.) 

Stillwater: Any reach in a stream of such level inclination as to have scarcely 
any perceptible velocity of flow; a sluggish stream, the water of which 
appears to be quiet or still. Local in Northeast. 

Strand: The shore or beach of the ocean or a large lake. 

Strait: A relatively narrow body of water connecting two larger bodies. 

Strath: A river valley of considerable size. (Scot.) 

Stream: Any body of flowing water. It may be of small volume, as a rill, 
great as the Mississippi or mighty as the Gulf Stream in the Atlantic 
Ocean. 

Stream Channel: The trench or depression washed in the surface of the 
earth by running water; a wash, arroyo or coulee. 

Stepto: An island in a lava field. Local in Northwest. 

Sugarloaf: A conical hill comparatively bare of timber. Local in far West. 

Summit: The highest point of any undulating land, as of a rolling plain, 
a mountain or a gap or pass in a mountain. 

Swale: A slight, marshy depression in generally level land. 

Swamp: A tract of Stillwater abounding in certain species of trees and coarse 
grass or boggy protuberances. 

Table: An elevated comparatively level bit of land between two streams. 
Local in Northwest. 

Table Land: A mesa. 

Table Mountain: A mountain having comparatively flat summit and one 
or more precipitous sides. A mesa. 

Talus: A collection of fallen disintegrated material which has formed a 
slope at the foot of a steeper declivity. 

Tank: A pool or water-hole in a wash. Local in arid West. 

Tarai: Extensive low-lying land in a river valley, partly swampy and covered 
with rank jungle. (East Indian.) 

Tarn: A small mountain lake. (Icelandic origin.) 

Terminal Moraine: A moraine formed across the course of a glacier, irregu¬ 
larly ridge-like in shape. 

Terrace: A relatively narrow level plain or bench on the side of a slope and 
terminating in a short declivity. 

Terrain: See Terrane. 

Terrane: An extent of ground or territory; a portion of the surface of the 
earth; the land. (Fr.) 

Terrene: Pertaining to the earth. (Fr.) 

Teton: A rocky mountain-crest of rugged aspect. Local in Northwest. 

Thalweg: A watercourse; a valley bottom; the deepest line or part of a 
valley sloping in one direction. (Ger.) 


GLOSSARY OF TOPOGRAPHIC FORMS. 145^ 

Thorofa' e : A level passage between lakes. The principal channel among 
islands. A highway. 

Tidal Marsh or Flat: Any marsh or flatland which is wetted by a tidal 
stream or sea. 

Tit : A small rocky protuberance on a hilltop. 

Tombolo : An off-shore island tied to the mainland by sand bars. (It. 
origin.) 

Tongue : A narrow cape. 

Tote-road : A road in the woods for hauling camp supplies. 

Tower : A peak rising with precipitous slopes from an elevated table land, 
Local in Northwest. 

Tributary: An affluent flowing into a large stream. 

Tundra : An upland or alpine marsh, the ground beneath which is frozen. 
There are great areas of tundra in the Arctic. (Rus.) 

Upland: A highland. 

Valley: A depression in the land surface generally elongated and usually 
containing a stream. 

Vlei: A vly. 

Vly: A small swamp, usually open and containing a pond. Local in Middle 
Atlantic States. (Dutch origin.) 

Voe: An inlet, bay or creek. (Icelandic origin.) 

Volcano: A mountain which has been built up by the materials forced from 
the interior of the earth, piling about the hole from which they were 
ejected. These may be lava, cinders or dust. 

Volcanic Neck: The solid material which has filled the throat or vent of 
a volcano, and has resisted degradation better than the mass of the 
mountain. It thus finally stands alone as a column or crag of igneous 
rock. 

Wash: The broad, dry bed of a stream; a dry stream channel. Local in 
arid West. 

Waterfall: Any single cataract. Both the terms waterfall and cataract 
may be applied to falls of like magnitude. 

Water Gap: A gap through a mountain occupied by an existing stream. 

Watershed: The ridge of high land or summit separating two drainage basins; 
the summit of land from which water divides or flows in two or more 

directions. 

The area drained by a stream. 

Well: Any excavation in soil or rock which taps underground water. 

Wind Gap: An elevated gap not occupied by a watercourse. 


PART II. 


PLANE AND TACHYMETRIC SURVEYING. 


CHAPTER VII. 

PLANE-TABLES AND ALIDADES. 

52. Plane and Topographic Surveying — Plane surveying 
consists of the representation of any portion of the surface of 
the earth in horizontal plan as it would appear viewed from 
vertical positions over every point on the surface. The result¬ 
ing map may be considered as consisting of an infinite number 
of points, the positions and relations of only so many of which 
are established as may be necessary to define the features 
which it is desired to represent, and this constitutes the in¬ 
strumental work of the survey. The prime element of posi¬ 
tion in the construction of such a map is a point in space. 
Such a position is indefinite, however, and to introduce 
the definite elements of direction and distance it is necessarv 
to add at least one other point. The addition of a third 
point introduces trigonometric functions by which any three 
elements of a triangle, except the three angles alone, serve to 
determine all others. These trigonometric functions may be 
solved or determined mathematically in figures from linear 
measures or angles, and graphically by means of the plane- 
table. 

The element of direction or azimuth is the deflection from 

146 



PLANE SURVEYING. 


14 7 


\ 


a true north and south line, or it may be a compass bearing, 
which is a deflection from the magnetic north and south line, 
or it may be the amount of deflection from any assumed 
line. The amount of such deflection of one line from another 
is measured by the angle formed at their intersection. The 
element of distance is measured by any unit conventionally 
established, the most definite of which in present use are the 
yard and meter (Art. 293). 

The representation of any portion of the earth’s surface 
by a topographic 7 nap requires that, in addition to the projec¬ 
tion upon a horizontal plane of a sufficient number of points 
to reproduce in plan the surface of the land, there shall also 
be indicated in some way the relief of the surface or its 
changes of height above, within, or below a fixed level surface. 
Such representation is made by determining instrumentally 
the elevation of such a number of points that the survey may 
be completed by drawing in the details between them. The 
new method of topographic surveying is by the determination 
of the least number of such points, and therefore calls for the 
greatest display of artistic and topographic skill, perception, 
and judgment on the part of the topographer. 

53. Plane-table Surveying. —The plane-table (Art. 56) is 
peculiarly well adapted to the mapping of topography, not 
only because it permits of quickly and graphically obtaining 
all the instrumental data which is requisite, but also because 
it has the added advantages, 

First, of having the map made in the field while the ter- 
rane is in view of the topographer; 

Second, the topographer can see at all times whether he 
has obtained all data necessary for the representation of the 
country; and 

Third, any insufficiency of instrumental or interpolated 
data can at once be supplied before leaving the field. 

In surveying ivith the plane-table the errors in measure¬ 
ment of horizontal angles can be so far eliminated in practice 


I4& PLANE-TABLES AND ALIDADES. 

that they may be neglected. In practice the horizontal pro¬ 
jections of existing angles are recorded graphically and are 
therefore free from errors of record, adjustment, or platting. 
In using the plane-table a number of points may have had 
their positions previously determined and platted on the map 
sheet. After the plane-table has been oriented and clamped 
each of these should be sighted from the first position occu¬ 
pied, and all other points in view should appear in vertical 
planes passing through the station and corresponding points 
on the sheet. Such points as do not meet this geometric test 
should be rejected until corrected or relocated graphically. 

In locating from a given station positions which are to be 
used in controlling the details of the sketching, a series of 
radial lines (Fig. 46) should be drawn from the station in all 




Fig. 46. —Drawing Radial Sight Lines. 


directions to salient points. This operation should be re¬ 
peated at other stations, and the intersection (Fig. 47) of any 
two on the same object gives its elementary location, a third 
line through the same point placing beyond doubt the accu¬ 
racy of its location. Accordingly, in this mode of surveying, 
constant opportunities occur for checking locations without 
calculation. The causes of failure to check may be immediately 






PLANE- TABLE SUP VE YING. 


I49 


tested, and the scale being ever present, it controls the amount 
of detail which it is necessary to gather. 

Another advantage of the plane-table as a surveying in¬ 
strument lies in the fact that, any two points being taken on 
the sheet as a base, a map may be constructed therefrom 
independent of scale, yet perfect in its proportion, by the 
method of intersection alone. If the base chosen be a very 
long one, as the two points of a trigonometric survey, and 

\2 /« 


'J 



A 

Fig, 47. —Intersecting on Radial Lines. 

the details of the plane-table survey be included within the 
limits of the base, then the survey is a contracting one with 
a diminishing chance of error, and each pair of intersections 
which has been tested geometrically becomes in -turn a base 
for further triangulation. Ultimately the length and azimuth 
of some line in this survey may be determined and the whole 
plane-table survey thus be reduced after the completion of the 
field work to any desired map scale. 

1-4. Reconnaissance and Execution of Plane-table Tri¬ 
angula tion. —Having located on the plane-table sheet (Art. 
188) two intervisible and well-defined points, the topographer 
should visit one of these and erect a signal of sufficient size 
to be visible from the most distant part of the territory cor- 









150 


PLANE-TABLES AND ALIDADES. 


responding to the plane-table sheet. Then selecting a num¬ 
ber of points visible from his position which may furnish 
satisfactory stations, a hasty reconnaissance trip is made over 
the territory, covering it all, if convenient, in the first recon¬ 
naissance, or perhaps only a portion, and returning to the 
reconnaissance after the plane-table work has caught up with it. 

This reconnaissance should consist in the selection of a 
few commanding and well-distributed stations, preferably on 
the highest eminences in the region under survey, and on 
each of these a signal must be erected on the point from 
which the greatest command of the surrounding country may 
be had. The distance apart of such secondary stations is 
chiefly dependent on the character of the topography and 
upon the scale of the map, and may be such as to correspond 
for the scale of the map chosen to a distance on the paper of 
five or six inches. In the course of this reconnaissance, a 
number of other stations may be selected by merely noting 
their positions and appearance, these being on prominent 
cleared points, such as bare rocks on mountain summits or 
slopes, a high lone tree, a building in a field, etc., or a few 
signal-flags may be placed, providing a sufficient number of 
such objects are not discovered. 

The topographer then begins his plane-table triangulation 
by occupying one of the two primary points and orienting on 
the other, lines being drawn to the secondary stations and to 
such other possible tertiary points as may be easily recog¬ 
nized (Fig. i). He then occupies the second of the primary sta¬ 
tions and orients from the first, intersecting on-such of the flags 
established as are visible from his position. Or, he may make 
his second station on one of the points sighted from the first, 
determining his position by resecting (Art. 74) from the sec¬ 
ond primary station. He continues thus until he has carried 
a secondary or skeleton triangulation over the entire area under 
survey, taking care not to spend too much time in sighting 
and attempting to locate minor and unimportant objects, but 


P LA NE - 7'ABI.E 7 PI A NG U LA TION. I 5 I 

devoting his attention primarily to the location of his second¬ 
ary stations and such other prominent objects as are readily 
recognizable and as may be of service in the further conduct 
of the plane-tabling. 

To this skeleton scheme of triangulation the topographer 
adjusts the traverse lines and the level elevations (Arts. 80 
and 129) and then proceeds to fill in the details of his map 
by further triangulation and the sketching of topographic 
forms, which should progress together hand in hand (Art. 13). 

55. Tertiary Triangulation from Topographic Sketch 
Points. —While the topographer is executing this outline tri¬ 
angulation, one or more assistants may be engaged in travers¬ 
ing roads and running lines of careful levels (Arts, 80 and 129) 
over the area under survey, so planning the latter as to well 
distribute the primary elevations to which levels of less accu¬ 
racy may be tied and adjusted. On the completion of the 
secondary or skeleton outline of triangulation, the topog¬ 
rapher will then have at his command a network of instru¬ 
mental control to which to tie future instrumental and topo¬ 
graphic details. He should adjust his traverses to his located 
points, add the elevations obtained by leveling or by vertical 
triangulation, and this network of control he proceeds to fill 
in with the topographic details (Arts. 13 and 15). 

As this filling in of details progresses he occasionally occu¬ 
pies tertiary plane-table stations, which should now be selected 
preferably at elevations midway of the slopes of the country, 
or, in other words, on lower elevations than the main plane- 
table stations, as in a vacant field on a hillside, a point on a 
road, or on alow bare summit. From such points he should 
not only draw lines and obtain intersections for the location 
of additional objects, as road intersections, buildings, hill¬ 
tops, etc., to which vertical angles must also be measured, 
but he should also sketch in as much of the detail of the 
topography as is clearly visible from his position and as is 
satisfactorily controlled by locations already obtained. 



152 


PLANE-TABLES AND ALIDADES. 


The above method of conducting a plane-table survey is 
not always practicable of execution. Frequently the country 
is unfavorable for the conducting of plane-table triangulation, 
owing to its being too wooded for the occupation of many 
stations, or because it is of such a generally uniform level as 
to offer few salient objects which may be located by triangu¬ 
lation. In such cases other methods must be employed to 
.fill in the topographic details, chiefly traversing of various 
kinds (Art. 80), but in any event it is desirable when practi¬ 
cable to control and tie these in by a skeleton plane-table tri¬ 
angulation executed between a few scattered points, which 
may be prepared for such triangulation by clearing, or by the 
erection of signals even at considerable expense of time and 
money (Art. 243). 

56. Varieties of Plane-tables. —The plane-table may be 
divided into four parts : 

1. The tripod ; 

2. The movements; 

3. The plane-table board; 

4. The alidade. 

There are in use in this country three general types of 
plane-tables, which, in the order of their rigidity and delicacy 
of mechanism, may be classified as follows : 

1. The Coast Survey plane-table ; 

2. The Johnson plane-table ; 

3. The Gannett plane-table. 

The first two differ little in the form of tripod and tripod 
3 egs, plane-table board and alidade, and greatly in the move¬ 
ment. The latter differs in all respects from the other two 
:and is adapted only to crude or rough reconnaissance work. 
To these may be added a fourth type, little used but of some 
value in exploratory or geographic surveys extending over a 
>Iarge scope of rough country. This is the folding plane-table, 
which has been chiefly used in the rough map-work of the 
Powell Survey and the early Geological Surveys. Its chief ad- 


PLANE-TABLE TRIPODS AND BOARDS. 


153 

vantages are its extreme portability, the tripod and board 
folding so as to occupy the least space. To the above 
instruments may still be added a fifth, also of the rougher 
reconnaissance kind ; namely, the English Cavalry sketching 
board, which is rarely used with a tripod, being attached to 
the wrist, but oriented and used as a plane-table; this in¬ 
strument is sometimes erected on a Jacob’s- staff. 

57. Plane-table Tripods and Boards. —The tripod legs 
of the Coast Survey and Johnson plane-tables (Figs. 48 and 
49), which are the only two forms adapted to accurate work, 
are as lightly made of wood as is consistent with requisite 
strength, shod with brass, and at the tripod head are of suffi¬ 
cient width to reduce lateral motion to a minimum. 

The plane-table board is made of well-seasoned pine, 
paneled with the grain at right angles, or more usually with 
a binding strip of wood dovetailed on its two ends at right 
angles to the grain, so as to counteract as much as possible 
the tendency to warp. The upper surface should be finished 
as nearly as possible in a plane, and when attached to the 
movement this surface should always be as nearly parallel as 
possible to the plane of revolution of the movement, so 
that the two planes shall remain parallel in all positions of 
the board. 

58. Plane-table Movements. —The movements of both 
forms of plane-table are of different construction, but both 
are of brass. Their design involves the same essentials, 
namely, sufficient strength for solidity ; horizontal revolving 
faces of large enough diameter and sufficiently accurate fit to 
prevent vertical motion when clamped together; and a means 
of clamping the axis of revolution to the tripod head when 
the revolving faces have been made horizontal by the leveling 
apparatus. 

The details of the Coast Survey plane-table are well illus¬ 
trated in Fig. 48, from which it will be observed that the tri¬ 
pod legs are split, widely separated, and attached to the tripod 



154 


PLANE-TABLES AND ALIDADES. 



F IG . 48.—Coast Survey Plane-table. 










































PLANE - TABLE MO VEMENTS. 


155 


head by binding-screws and clamps; that the movement is at¬ 
tached to the tripod head by means of a center clamping- 
screw as in ordinary surveying-instruments; also that the 
leveling is effected by the usual form of three leveling-screws; 
and that the horizontal motion is obtained by two heavy 



Fig. 49. —Telescopic Alidade and Johnson Plane-table. 

circular plates sliding one upon the other, the lower attached 
to the tripod head by the center clamping-screw, and the upper 
to the plane-table board by clamping-screws, and both fast¬ 
ened together by a center axis of revolution. There is in ad¬ 
dition a clamping-screw for fixing the instrument in orienta¬ 
tion, and a tangent screw for slow motion. It will be observed 

















PLANE-TABLES AND ALIDADES. 


1 56 

that this instrument is very heavy, is rather difficult of ma¬ 
nipulation because of inaccessibility of leveling and clamping- 
screws, and is in fact too cumbersome for convenient use, ex¬ 
cepting where travel is easy. 

The Johnson plane-table (Fig. 49), so named after its in¬ 
ventor, Mr. Willard D. Johnson, is used by the United States 
Geological Survey, and though not quite as rigid as the Coast 
Survey type, is sufficiently so for all practical purposes and is 
much lighter, more portable, and more easily manipulated. 
The movement is also more compact and less liable to de¬ 
rangement or injury. It consists of a split tripod securely 
attached to the head as in the case of the Coast Survey tripod, 
but the leveling and horizontal movements are entirely unique 
in surveying-instruments, being essentially an adaptation of 
the ball-and-socket principle, so made as to furnish the largest 
practicable amount of bearing surface. 

They consist of two cups, one inside the other, the inner 
surface of the one and the outer surface of the other being 

; 


a. Plane-table Board /. Upper Level Cup 

b. Bearing Plate g. Lower “ “ 

c. Tripod Head h. Level Clamp 

d. * “ Legs i. Azimuth Clamp 

e. Azimuth Cup 

Fig. 50.—Johnson Plane-table Movement. 

ground to fit as accurately as possible. The interior cup con¬ 
sists of two parts or rings, one outside the other, one control¬ 
ling the movement in level, and the other that in azimuth (Fig. 
50). From each of these there projects beneath the move¬ 
ment a screw, and each screw is clamped by a wing nut. 
These cups and rings are bound together and to the tripod 















TELESCOPIC ALIDADES. 


i5 7 


head by the two nuts, and are attached to the plane-table 
board by screwing it over a center axis or pin projecting from 
the upper surface of the upper cup. The instrument is first 
leveled, not by leveling-screws, but by the ball-and-socket mo¬ 
tion given by the pair of cups which are clamped by the 
upper screw when the board is level, the latter being still left 
free to revolve horizontally for orientation and being clamped 
by the lower screw. There is no tangent screw for slow mo¬ 
tion in azimuth, it being possible owing to the long lever-arm 
furnished by the outer edges of the board to move it with 
sufficient slowness for all practicable purposes. 

59. Telescopic Alidades. —Alidades used with the more 
rigid plane-tables differ in form according to the character of 
the work to be executed. Where the instrument is used 
chiefly in triangulation, the alidade should be of the most 
approved type and the rule should be of sufficient length to 
permit its being used as a straight-edge in drawing lines from 
one extreme of the board to the other. In practice this 
rarely exceeds 25 inches in length. In using the plane-table 
in sketching or traversing a smaller alidade and one having a 
straight-edge not exceeding eighteen inches in length will be 
found more portable and better adapted to the work re¬ 
quired. 

The alidade generally employed by the U. S. Coast and 
Geodetic Survey consists of a brass or steel straight-edge 
2 2 by 3 inches in width and 12 to 15 inches long, from and 
perpendicular to which rises a brass column 3 inches in height, 
surmounted by Y’s in which rest the transverse axes of the 
telescope. To one end of the axis is firmly attached an arm 
of thirty degrees, graduated to minutes on either side of a 
central zero, the accompanying vernier being attached to the 
Y support. On the telescope-tube are turned two shoulders 
on which rests a striding-level. There is a clamp and tangent 
screw for slow motion for moving the telescope in vertical arc, 
and on the straight-edge are two small spirit-levels at right 


i 5 8 


PLANE-TABLES AND ALIDADES. 


angles to each other. A declinatoire accompanies the alidade 
and is carried in a separate box or is sometimes attached as a 
part of the striding-level. The declinatoire box is oblong, 
with the sides parallel to the north and south lines and grad¬ 
uated to about 5 minutes on either side of the zero. 

The chief difference between this alidade and the one 
used by the United States Geological Survey (Fig. 49) is that 
the straight-edge of the latter is 18 or 24 inches in length, 
with one edge beveled and graduated to the scale of map- 
work. The telescope is on a standard 4 inches high, has a 
focal distance of 15 inches and a power of 20 diameters, with 
an objective of if inches diameter. The telescope revolves 
horizontally in a sleeve, with a stop for adjustment of vertical 
collimation. Instead of two small levels attached to the 
straight-edge, a single detached circular level is carried by 
the topographer. 

The smaller telescopic alidade used by the United States 
Geological Survey (Fig. 51) on traverse and stadia work is 



Fig. 51.—Telescopic Alidade. 


more like the Coast Survey alidade in having a shorter tele¬ 
scope and focal distance and a shorter straight-edge. The 
vertical arc, instead of being graduated on the side and read¬ 
ing against a vernier as is customary with other surveying in- 























































ADJUSTMENTS OF TELESCOPIC ALIDADE. I 59 

struments, is a sector, which, instead of pointing downwards, 
points towards the rear or eyepiece of the telescope and is 
graduated on its outer surface. This is read against a vernier 
fixed in such position that the reading may be made from one 
position of the observer at the eyepiece without his moving 
to the side of the plane-table as in ordinary instruments. 

60. Adjustments of Telescopic Alidade. —There are 
practically no adjustments to the plane-table and alidade 
excepting the adjustments of the latter for striding-level and 
collimation. Adjustments or tests may be made of the 
straightness of the fiducial edge of the rule by drawing a line 
along it, and reversing it, placing the opposite ends upon the 
marked points and again drawing a line; if the two lines do 
not coincide, the edge is not true. It is not necessary that 
the two edges of the straight-edge be exactly parallel, if care 
is always taken in using the instrument to draw along but 
one edge. 

Attached levels when used may be adjusted by placing the 
alidade in the middle of the table, marking its edges on the 
paper, and bringing the bubble to center by means of the 
leveling apparatus; then it is reversed 180 degrees, and if the 
bubble be not in the center it is corrected one-half by level¬ 
ing the table, and the other half by adjusting the screws of the 
attached levels. 

The striding-level is adjusted by placing the alidade in the 
center of the table, leveling the telescope by the vertical tan¬ 
gent screw, then reversing the level upon the telescope. If 
the bubble come not to the exact center of the tube, half of 
the error is to be adjusted by the screws in the level, and the 
other half by releveling the telescope with the tangent screw. 

In addition to the above there are a few other adjustments, 
as that of making the line of collimation perpendicular to the 
axis of revolution of the telescope, and of making the latter 
parallel to the plane of the rule, and for parallax, and to cor¬ 
rect the zero of vertical arc, etc. None of these need, how- 


l6o PLANE-TABLES AND ALIDADES. 

ever, be described, as in the alidade as now made the bearings 
of the axes are unchangeable and there is either a means of 
setting the vernier at zero or an index error is to be read. 

61. Gannett Plane-table. —Where rough traverses are run 
in connection with the making of small-scale maps, and a firm 
board is unnecessary since the telescopic alidade is not used, 
an exceedingly convenient and portable plane-table is that 
employed in the U. S. Geological Survey and known as the 
Gannett plane-table (Fig. 52), after its originator, Mr. Henry 



Fig. 52.—Gannett Traverse Plane-table and Sight Alidade. 

Gannett. The tripod of this instrument is very light, consist¬ 
ing of three straight legs made of single pieces of wood. 
These are shod with metal tips and attached by bolts and nuts 
to the head, which is a simple plate 3^ inches in diameter. 
The board, which is 15 inches square by f of an inch in thick¬ 
ness, is a well-seasoned piece of pine and is attached to tl e 
tripod by a single center screw. There is no leveling apparatus, 
the instrument being leveled by means of the tripod legs, and 
there is no means of clamping the instrument in azimuth, the 
movement in azimuth being controlled by friction, and the 
board being held in place by friction due to the tightness with 














SIGHT-A LID AD ES. 


1 6 1 


which it is bound to the tripod head by the center screw. 
In running traverses the table is not oriented by backsights 
and foresights, but is adjusted in azimuth by means of a com¬ 
pass needle or declinatoire having a range of 5 0 to 8° and 
placed in a small oblong box 3 to 5 inches in length set into 
the side of the board. 

62. Sight-alidades. —The alidade used with the Gannett 
plane-table (Figs. 52 and 53) is 6 inches in length, o. 1 inch 
in thickness, and 0.6 of an inch in width, of brass, with a fold¬ 
ing front sight with vertical hair 3^ inches in length, with V 
sight-notch in the top and a short peep-sight in the rear. 
The fiducial edge is beveled and graduated to the scale of 
the map. 

To determine elevations of near objects in traversing with 
light traverse outfit, a small sight-alidade was devised by the 
author both for sighting directions and for determining eleva¬ 
tions by vertical angulation. (Fig. 53.) This consists of a 



ruler nearly 7 inches in length, f of an inch wide, and T \ of an 
inch thick, made of brass with a beveled fiducial edge divided 
to hundredths of a mile on the scale of the field-work. At the 
rear end is a fixed sight one-half inch high with a notched 
gun-sight, and beneath this is a fine peep-sight. At the far 
end is a hinged sight nearly 3 inches in height, a little over 

































16 2 


PLANE-TABLES AND ALIDADES. 


one-half inch in width, and with a slot T l 7 of an inch in width 
extending nearly its entire length. 

So far it is quite similar to the ordinary sight-alidade used 
in traversing only. It differs from this in having in addition 
a small level-bubble attached to it near the rear end, and close 
to this is a leveling-screw with milled head. The forward 
sight has ruled on it a tangential scale on which the smallest 
division is equivalent to 20 feet vertical elevation at the unit 
distance of 1 mile. Running on this is a slide with horizontal 
cross-hair, and the traverseman in sighting any object applies 
his eye to the peep-sight if the object is above, or to the slid¬ 
ing scale on the hinged sight if the object is below him, and 
moves the slide up and down until the horizontal cross-hair is 
in contact with the top of the object sighted. He then notes 
the reading on the tangent scale, and measuring on his trav¬ 
erse-board in hundredths of miles the distance from his occu¬ 
pied point to the point sighted, he multiplies the reading of 
the scale by this distance, and the product is the difference in 
height in feet. The alidade must necessarily be leveled by 
the small milled-head screw or by the plane-table movement 
at each sight taken, and the position of the object sighted is 
determined, in case of traverse, by intersection from various 
traverse stations. 

This instrument is most used in rough traversing and 
in sketching topography where there are no roads and it is 
imposssible to carry heavy instruments, because the traverse- 
man moves either on foot or horseback; its use is limited in 
distance, but for work on a scale of one or two miles to the 
inch it gives comparatively accurate results for distances not 
exceeding one or two miles and for elevations less than 8 
or 10 degrees. In sketching in details of topography 
along a road traveled on foot or by conveyance it is also 
convenient in determining the elevations of unimportant 
points near by, as it is much more rapidly used than the tele¬ 
scopic alidade. 


FOLDING EXPLORATORY PLANE-TABLE . 


163 


63. Folding Exploratory Plane-table. —This consists of 
a folding split-leg tripod similar to those made for supporting 
photographic cameras, but a little more substantial. The 
three legs are carried in a canvas case 24 inches in length and 
3 by 4 inches in cross-section. The tripod head consists of a 



Fig. 54. —Folding Exploratory Plane-table and Small Theodolite. 

triangular block of wood 7 inches on each side by 1 inch 
thick, with metal pegs on the under side into which the split 
legs of the tripod are sprung, and carrying a centre binding- 
screw for clamping the plane-table movement. This latter 
consists of three small bronze arms, in general shape like 
those of a theodolite or transit, supported by three leveling- 








































PLANE-TABLES AND ALIDADES. 


164 

screws and having a clamp and tangent screw. (Fig. 54.) 
The top of the movement is a screw 3-J inches in diameter, to 
which are fastened the cross-braces which support the board. 
These are two strips of wood, 24 inches in length by 3 inches 
in width and 1 inch in thickness, and to the four ends of 
these cross-arms are screwed the outer slats of the folding 
board. 

The plane-table board consists of 24 wooden slats, each 24 
inches in length, 1 inch in width, and J inch in thickness, and 
bound together by heavy canvas glued to one surface in such 
manner that the whole can be rolled into a compact, cylin¬ 
drical form and carried in a case 24 by 6 inches in diameter 
or be kept unrolled and clamped to the binding-strips. The 
surface of this plane-table board is so uneven that good work 
cannot be carried over any considerable area without appreci¬ 
able error. Accordingly, there is used in conjunction with 
this instrument a small theodolite with 5-inch circle, which is 
screwed to the plane-table movement in place of the board. 
(Fig. 54.) The alidade used with this instrument consists of 
a simple straight-edge of brass, 18 inches in length, with fold- 
ing sights, the foresight being a slot with two or three peep¬ 
holes. With this apparatus the writer has carried a system 
of plane-table triangulation, accompanied by vertical and 
horizontal angles with the gradienter, and has made a com¬ 
plete geographic map on a scale of 4 miles to 1 inch and with 
sketched contours of 200 feet interval, in a season of seven 
months, over an area of 11,000 square miles. The extreme 
error of location on the plane-table, as afterwards corrected 
by the gradienter angles platted to a primary theodolite tri¬ 
angulation, was a little in excess of J inch in a linear distance 
of 140 miles. 

64. Cavalry Sketch-board. —This is a modified plane- 
table devised by Captain Willoughby Verner of the British 
Army. It has an extreme length of 9 to 12 inches and an 
extreme width of 7 to 9 inches. (Fig. 55.) On either side 


ROLLER 


CA VALR Y SKETCH-BOARD 


16 


5 












































































































1 66 PLANE-TABLES AND ALIDADES. 

are two rollers held by friction thumb-screws over which a 
continuous roll of paper is passed. At one end of the board 
is a declinatoire or small box compass, while on its under side 
is a pivoted strap by which it can be fastened to the wrist of 
the surveyor and revolved for orientation. This apparatus is 
used chiefly as a traverse plane-table board, the line of direc¬ 
tion through the compass being parallel to the general direc¬ 
tion of the route traversed. An attachment to the under side 
of the board permits of its being fastened to a light tripod or 
Jacob’s-staff, when desired. An adjunct to its use is a light 
alidade with scale, and it is employed much as is a plane- 
table excepting that its range is limited by the angle seen 
ahead when attached to the wrist. Instead of a declinatoire 
a small magnetic compass may be counter-sunk in a collar in 
which it can be revolved, and on the glass of the compass a 
fine line is engraved which is termed the working meridian. 

To use the board the working meridian is set in the direc¬ 
tion in which the traverse is being run by turning the compass- 
box around in its socket, the relative positions of the working 
meridian and the board being thus determined. The latter is 
set for sketching by revolving it on the strap pivot until the 
working meridian coincides with the magnetic needle when at 
rest. In order to prevent the paper rollers from working 
loose, a thumb-screw is provided on the end of either roller 
by which it can be clamped so as to regulate the degree of 
friction with which it moves. If a half-circle protractor is 
attached to the side of the board with a plumb-line suspended 
therefrom, the angle of a slope may be determined by sight¬ 
ing along the edge of the board held on edge and reading 
the position of the plumb-line on the protractor as a slope 
board. 

This cavalry sketching-board will prove most serviceable 
in running rough meanders or traverses, and when held in 
the hand may be used with as much accuracy as the prismatic 


BATSON SKETCHING-CASE. 


166 a 


compass, while there is added to the process of eye-sketch¬ 
ing all the data which can be incorporated on a plane-table 
sketch. 

a. Batson Sketching-case.— This is an American modifi¬ 
cation of the Cavalry Sketch-board and is used generally by 
the army. It differs chiefly from the sketch-board in having 
upon its upper surface a movable graduated circle which carries 
a small alidade with scales. The compass at one end is a 
declinatoire of small angular range, like that on the Gannett 



Fig. 55a,—B atson Sketching-case. 


plane table. Near it a clinometer is let into the case. Six 
holes in the edge of the board opposite the compass are for use 
as oencil holders. 

JL 

The protractor is held in position by a carrier which slides 
upon a bar attached to the wooden end-pieces. The protractor 
can be turned or clamped by two set screws. It can also be 
lifted off the paper which passes under it by pulling back a 
spring catch at the end of the carrier bar. The alidade turns 
within the graduated circle and with it forms the protractor. 
The paper used is six inches wide and twenty to forty inches 
in length. 












166 b 


PLANE-TABLES AND ALIDADES. 


To mount the paper, raise the protractor to a vertical posi¬ 
tion, insert one end of the paper in the slit of the far roller and 
turn the roller until only seven or eight inches of the paper are 
left free, then insert the free end in the slit of the near roller 
and turn the roller until the paper is taut. Turn the protractor 
down on the board. 

To set the instrument release the needle. Face in the 
general direction of the route to be mapped and hold the in¬ 
strument in the left hand with the compass to the right ; orient 
it until the long way of the paper is in the general direction of 
the route to be mapped. Read the bearing, unclamp the pro¬ 
tractor and turn it until the index on the upper plate of the 
carrier indicates the same reading on the protractor that is 
shown by the needle. Move the paper by turning one of the 
rollers until the point occupied is opposite the centre of the 
protractor. Unclamp the carrier of the protractor and slide it 
along the bar until the centre of the protractor is over the 
point. 

Select a point on the paper for the initial station, preferably 
about midway of the edges of the paper. Now, holding the 
instrument in the left hand, or having it mounted on the staff, 
orient it until the reading of the needle corresponds to the 
reading of the protractor circle. Turn the alidade upon all 
objects it is desired to plot on the map, and draw light lines 
in the slit from the station-point toward them. After having 
taken the slopes in the vicinity and sketched in the detail, the 
ruler is clamped in the direction of the next station and a 
sight line drawn. 


CHAPTER VIII. 


■# 


SCALES. PLANE-TABLE PAPER, AND PENCILS. 

65. Scales. —The scales of topographic map are of two 
kinds or dimensions, horizontal and vertical. The horizontal 
scale is the ratio obtained by dividing the distance between 
two points on the map by the corresponding distance on the 
ground, both being reduced to a common unit, generally 
inches. Scales are expressed in three ways. 

1. As ratio scales ; thus - or 1 : 1200 means that one 

1200 

inch, centimeter, etc., on the map represents 1200 inches, 
centimeters, etc., on the ground. Or, as there are 12 inches, 
to one foot, one inch on the map represents 100 feet on the 
ground. 

2. Scale may be graphically expressed by drawing a line 
on the map and dividing it into equal parts, each division re¬ 
presenting and being marked with the corresponding distance 
on the ground. Thus if the scale is one inch equal to 100 feet, 
divide an inch on the map into tenths and hundredths, when 
one-tenth of an inch on the map will represent ten feet on the 
ground. 

3. A statement in words may be used to represent the 
scale ; thus, 1 inch = 1200 feet or 1 inch = 2 miles. 

The scale of a map is selected according to the use it is to 
serve. If the map is of a farm or a reservoir site or a railway 
alignment, and minor details of fence or property lines and 
plans for construction are to be shown upon it, a large scale,, 

166c 



166 d SCALES, PLANE-TABLE PA PEP, AND PENCILS . 


as i inch = ioo feet or i inch = 500 feet, should be adopted. If 

to show only details of topography in a well inhabited country 

* 

a scale of 1 inch= 1 mile more or less, will best serve the pur¬ 
pose. If an exploratory or geographic map is desired a very 
small scale, as 1 inch= 10 miles more or less, will be best, as it 
will permit the depiction of the largest possible area of country 
over a map of given size. 

Vertical scale is represented on topographic maps by lines 
of equal elevation or contour lines (see Art. 192, p. 455). The 
scale or the interval between contour lines is selected accord¬ 
ing to the horizontal scale of the map or the amount of detail 
to be shown and the use to which the map is to be put; it also 
depends on the ruggedness or smoothness of the region 
mapped. For detailed maps having horizontal scales of 1 
inch =100 feet, etc., the contour interval should be from one 
to five feet. For topographic maps of horizontal scales about 
1 inch = 1 mile, the contour interval may be from five to fifty 
feet. Vertical scale is denoted by writing the figures of eleva¬ 
tion above some datum, as sea level, represented by the con¬ 
tour lines upon those lines. 

65a. Special Scales. —In the execution of any topographic 
survey some special scale is selected for the field platting. 
This may be 100 feet or 1000 feet to the inch, or 1 mile to 
the inch, etc. ; but be the scale whatever it mav, the work of 
platting distance will be greatly facilitated by the construction 
of special scales which will reduce the field measurement 
directly to relative distances on the map. As in Article 95, in 
which a special scale for reducing paces of men or animals or 
time of travel to map scale is shown, scales or tables should 
be constructed in odometer work, in which a certain number 
of revolutions of the wheel shall correspond directly with so 
many divisions of the platting scale (Art. 98). Such scales 
can be easily prepared by the topographer. They may be so 
divided that given distances on the scale represent so many 


SLIDE-RULE. 


167 


revolutions; or a mile or inch scale may be used and a table 
constructed in which a given number of revolutions for a 
given sized wheel will correspond to a fixed proportion of the 
mile or inch. 

In such topographic mapping as is executed by large 
organizations, as the U. S. Geological Survey, standard scales 
are adopted for field-work, as 1 : 48,000 for the larger-scale 
topographic maps and 1:96,000 for the smaller-scale maps, 
and boxwood or steel rules are obtained from the various 
makers on which a distance corresponding to a mile on a scale 
of 1:48,000 is divided into 100 parts. Then if the topog¬ 
rapher measures a given fraction of a mile with the odometer, 
chain, or stadia, he plats the same on the map, not by reducing 
it to inches (Art. 189), but by his scale of miles. Likewise 
for computing vertical angles he has but to measure the dis¬ 
tances between two points when the result is given him, not 
in inches, but in tenths and hundredths of a mile, and that 
quantity can be quickly computed by slide-rule (Art. 66) or 
table, since these are generally prepared for mile or foot 
measurements and not for inch measurements. 

Similar scales or diagrams greatly facilitate the work of 
platting triangulation points and projecting maps. A scale 
has been devised by Mr. A. H. Bumstead of the U. S. Geo^ 
logical Survey for platting projections and triangulation points 
on a scale of 1 : 48,000 or multiples thereof. This saves all 
the work of reducing the odd minutes and seconds between 
platted projection lines to distances on the map scale (Art. 
188), as the scale is divided into minutes and their fractions 
of latitude and longitude on the fixed map scale. Similar 
scales may be graduated for other map units. 

66 . Slide-rule. —The slide-rule consists of a number of 
scales which slide one on the other and are logarithms of 
numbers platted to scale. These scales are so arranged 
that the corresponding logarithms may be brought opposite 


1 68 SCALES, PLANE-TABLE PAPER , AND PENCILS . 


each other so as to mechanically add or subtract. By 
its use nearly all forms of multiplication and division, in¬ 
volution and evolution, including trigonometric operations and 
computations, may be performed. Slide-rules are made not 
only for the ordinary operations of multiplying and dividing, 
but also for special use in computing stadia measures and for 
computing engineering quantities of various kinds. 

A topographer who has much stadia work to compute and 
who has no specially prepared tables or diagrams for his scale 
of work should use a slide-rule. Likewise in computation of 
vertical angulation a slide-rule should be used where tables to 
scale are not at hand. The instrument performs accurately 
and without mental effort amass of tiresome calculations, mul¬ 
tiplications, and divi ions, which could not possibly be worked 
out by ordinary methods in nearly so rhort a time as by its 
use. Where accuracy L desired slide-rules may now be pro¬ 
cured made with the graduations on celluloid facings. As a 
result very fine readings can be made, especially as the brass 
runner of the older forms is superseded by a glass plate on 
which line lines are ruled. 

67. Using the Slide-rule.—The following simple explana¬ 
tions of the use of the slide-rule in such operations as the 
topographer has to perform are extracted from an article bv 
Mr. G. B. Snyder published in the Engineering News. For 
better understanding of this explanation the four scales 
on the slide-rule shown in Fig. 56 are marked A, B , C, 
and D. 

Multiplication .—To multiply, set the index of the slide 
opposite the multiplicand on the rule, the result will then 
be found on the rule under the multiplier on the slide. 

Example: Multiply 2 by 3. 

Using scales A and B, set the left index of B under 2 
on A, then over 3 on B will be found 6 on A, and all 


USING THE SLIDE-RULE. 


169 


the other numbers on B will be found to be in 



RIGHT INDEX 


LEPT INDEX 


Fig. 56. —Scales of the Slide Rule. 

proportion with those on A. Thus, 4 will 
unde: 0, 6 unde 12, 7 u der 14, etc., or, 


the same 


be found 
c onsidcred 

























































170 SCALES, PLANE-TABLE PA PEP, AND PENCILS. 

as a proportion, 1:2:: 3:6::4:8:: 6: 12, etc., or \ — | 
z= £■ = , etc. These results will be proportionally the same 
whatever value we assign to the numbers, which can be con¬ 
sidered as 200, 600, 800, etc., as 20, 60, 80, etc., or as .2, 
.6, .8, etc. 

Rules for the position of the decimal-point are given in the 
pamphlet that accompanies the rule, but usually its position 
can be obtained by inspection. 

The above multiplication can be performed on the lower 
scales, when finer readings can be obtained. If the left-hand 
index of C is set over 2 on D, it will be found that numbers 
above 5 on the slide protrude beyond the rule. To obtain 
these results the right-hand index of C must be set over 2 on 
D , when 12 will be found under 6, 14 under 7, 16 under 8, 
etc. 

Division .—The process of division is merely the reverse of 
multiplication. The divisor is set opposite the dividend, and 
opposite the index is found the quotient. 

Example: Divide 20 by 8. 

Using the lower scales, set 8 over 20; under the index will 
be found 2.5. 

Squares and Square Roots .—On scales A and B there are 
in the length of the rule two complete sets of numbers, while 
there is only one set of numbers on scales C and D, the num¬ 
bers on the lower scales taking up twice the distance they do 
on the upper. To square a number its logarithm must be 
multiplied by 2, and to obtain its square root its logarithm 
must be divided by 2, and as the distances on the rule repre¬ 
sent logarithms of the numbers affixed to it, the numbers on 
the upper scales are the squares of those on the lower. 

To square a number, set the runner to the number on the 
lower scale, and the coinciding number on the upper scales 
will be its square. Thus, over 2 will be found 4; over 5 will 
be found 25 ; over 15 will be found 225, etc. 

To obtain the square root the above operation is reversed 


USING THE SLIDE-RULE. 


171 

by setting the runner to the number on the upper scale; the 
coinciding number on the lower scales will be the square root. 
Thus, under 9 will be found 3, under 16 will be 4, under 625 
will be 25, etc. As there are two sets of figures on the upper 
scale, care must be taken that the proper one is used ; thus, 
in obtaining the square root of 9, the 9 on the left scale must 
be used, for if the runner is set to the 9 on the right-hand 
scale, its coincident number will be found to be 9.48 -J-, which 
is the square root of 90. 

If the number whose square root is to be taken has an odd 
number of figures in it, counting the figures in front of the 
decimal-point, use the left-hand scale; if an even number, use 
the right-hand scale. Thus, with 625 use the left-hand scale; 
with 62.5 use the right-hand. If the number is all decimal, 
use the right-hand scale. 

The Solution of Plane Triangles .—The under side of the 
slide is graduated to a scale of sines and a scale of tangents, 
so that trigonometric calculations can be made on the rule. 
When the under side of the slide is uppermost, the scale of 
sines will be along scale A and the scale of tangents along 
scale D. By referring to a table of natural sines, it will be 
found that 1.000 is the sine of 90°, that .100 is the sine of 
5° 44', and that .010 is the sine of about 0° 34^-', so the right 
index of the scale is 90°, the middle index is 5 0 44', and 
the left index is o° 34^. Sines of less than o° 34^' can be 
found by setting 34J- on B under the right index of A ; then 
over any number of minutes on B will be found the corre¬ 
sponding natural sine on A. Note the graduations on the 
scale of sines; as rules are usually graduated, every degree is 
marked between 40° and 70°. Above yo° the shorter marks 
are every 2° until the first long mark is reached, which is 8o°. 
There is only one mark (85°) between 8o° and the index. 

By referring to a table of natural tangents, 1.00 will 
be found to be the tangent of 45°, and .100 to be the 
tangent of 5 0 43', so the right index of the scale of tan- 


1 72 SCALES, PLANE-TABLE PAPER , PENCILS. 

gents is 45°, and the left index is 5 0 43'. To obtain 
tangents less than 5 0 43', set 5 0 43' = 5.72 on C, over 
the right index of D ; then under the angles expressed in de¬ 
grees and decimals on C will be found their corresponding 
natural tangents on D. With the slide set as above, the tan¬ 
gent of i° will be found to be .01745. To find the tangent 
of angles less than i°, set 60 opposite 1745, and minutes on 
the slide will be opposite their corresponding tangents on the 
rule. Tangents of angles greater than 45 0 can be obtained by 
dividing 1 by the tangent of the complement of the angle. 

Triangles can be solved very readily on the slide-rule and 
with considerable accuracy if not more than three or four fig¬ 
ures are necessary in the results. 

Right-angled Triangles. —Example: What is the altitude 
of a right-angled triangle, with an angle at the base of O 0 25' 
and a hypothenuse of 1240 ? Here the angle is smaller than 
can be read on the scale of sines. On the scale of sines will be 
found a mark for single minutes near the 2° mark. Set this 
mark to the under index of the rule, then minutes can 
be read along B and their corresponding sines will be found 
on A. 

As noted before, 34^ is about as low as can be read on 
the scale of sines. With the slide set as above, 34J' will 
nearly coincide with the index, and T will be found under 
.000291, which is its sine. Set the runner to 25' on B and 
move index to runner; over 1240 will be found 9.0. The po¬ 
sition of the decimal-point can be found by a mental calcula¬ 
tion, thus: As noted before, the indexes of scale A correspond 
with o° 34j\ 5° 44', and 90°, respectively; if the angle had 
been 90°, the altitude would have been 1240; if it had been 
5 0 44', the altitude would have been 124.0; if it had been 
o° 34j, the altitude would have been 12.40; the angle is 
o° 25', therefore the result must be less than 12.40. 

Example: Given a right-angled triangle with a base 64 ft. 
and an angle at the base of 42 0 3 1 / ; what is the altitude ? Set 



PLANE-TABLE PA PEP. 


i/3 


index over 64. Under 42 30' on the scale of tangents will 
be found 58.6 ft. 

Example : What is the altitude of a triangle with a base 
of 24.5 ft. and an angle at the base of 72 0 15'? 

H ere the angle is greater than 45°, and cannot be read on 
the scale of tangents, so the complement of the angle is used 
and divided into the base instead of multiplying. 90° — 
72 0 15 == 1 7 ° 45 - Set 17 0 45' over 24.5 ; under index will be 
found 76.5 ft., the altitude required. 

Owing to the ease with which numbers can be squared on 
the slide-rule, work can readily be checked by seeing if the 
square root of the sum of the squares of the two legs is equal 
to the hypothenuse. One of the simplest ways of avoiding 
mistakes is to bear in mind that sines and cosines are merely 
percentages of the hypothenuse, and that tangents and co¬ 
tangents are percentages of the base or altitude. 

Plane Triangles .—The preceding examples have been ap¬ 
plied to right-angled triangles only. The following are ap¬ 
plied to plane triangles in general: 

Example: Given one sicie and the angles of a triangle 
to obtain the remaining sides. 

Here we use the proposition: Sine of the angle opposite 
the given side : sine of the angle opposite the required 
side : : the given side : the required side. 

To solve the above problem, set 64°, the given angle, on 
scale of sines, under 117, the given side on A \ then over 
76° will be found 126.3, the length of its opposite side, and 
over 40° will be found 83.7, the length of its opposite side. 

With the three sides given and one of the angles the re¬ 
maining angles can be found in the same way, and with two 
sides given and the angle opposite to one of them, the solu¬ 
tion is equally simple. 

Example: Given a triangle with a side of 81 ft. and a side 
of 60 ft., with an opposite angle of 40°. Required the re¬ 
maining side and the remaining angles. 


174 SCALES, PLANE- TABLE PA PEP, AND PENCILS. 


Set 40° on scale of sines under 60 on scale A ; then under 
81 will be found 6o°, being the opposite angle, and the re¬ 
maining angle will be 180° — (6o° -f- 40°) = 8o°; over 8o° will 
be found 92, the remaining side, 

68 . Plane-table Paper. —In conducting an accurate plane- 
table survey the paper employed is as important an instru¬ 
ment and should be selected and handled with as great care 
as other portions of the outfit. An accurate scheme of plane- 
table triangulation cannot be developed and delicate inter¬ 
section obtained from lines drawn on inferior paper or on 
paper that presents an uneven surface. The practice of using 
large sheets of paper only a portion of which is attached to 
the board at one time, the remainder being rolled up and re¬ 
tained in position by clamps, is to be discouraged. The 
rolling of the paper produces cracks and causes it to buckle in 
such manner as to render it impossible to obtain the most 
satisfactory surface on which to rest the alidade. Moreover, 
the cumbersome roll at one or both ends of the board pre¬ 
sents a large surface to the winds and renders it difficult to 
keep the table steady from vibration even in winds of mod¬ 
erate velocity. Finally, paper is very sensitive to atmos¬ 
pheric changes; especially is it affected by the moisture in 
or dryness of the atmosphere, and points plotted twenty to 
thirty inches apart will frequently be found in error after a 
lapse of but a few days, and by a very appreciable amount if 
any but the best paper is used. 

The best plane-table paper is double-mounted, and is pre¬ 
pared in the following manner: A rectangular wooden frame 
a little larger than the size of the sheet required is made, and 
over it is tightly stretched, by means of tacks, a piece of the 
ordinary muslin or cotton cloth used in map-mounting. To 
each side of this is pasted, with the right surface out, a sheet 
of the best drawing-paper, so oriented that the grain of the 
two sheets will be crossed at right angles. The result is a 
sheet of “double-mounted” drawing-paper; one which is 


PREPARATION OF FIELD SHEETS. 


175 


least affected by atmospheric changes, and it has been found 
by experiment that such changes affect it almost uniformly in 
all directions. Therefore, if variations take place in its di¬ 
mensions, they are of such kind as may be largely eliminated 
by a uniform reduction or enlargement of scale. Such 
double-mounted paper can now be purchased of most of 
the larger dealers in drawing and surveying instruments, 
and the best paper for this purpose has been found to be 
paragon grade of heavy eggshell or double elephant paper. 
Such plane-table sheets cannot be rolled, and must be trans¬ 
ported in flat wooden boxes, or else be laid against the sur¬ 
face of the plane-table board and carried in a suitable canvas 
or leather case. 

For less important plane-table work, or for plane-tali le 
traverse, ordinary single-mounted drawing-paper of good quality 
may be used, and this may be rolled, or single sheets may be 
used and transfer made from one to another by long orienting 
marks on the board and on the paper. For plane-table work in 
a region of bright sunlight where the glare affects the eyes, it 
has been found desirable to use tinted drawing-paper in prefer¬ 
ence to plain white, and the most satisfactory are the neutral 
tints between Paine’s gray and slate-blue. Celluloid sheets 
having white or epaque surface are very useful in regions like 
the Adirondacks or the Northwest, where there is much rain 
and dew. Sheet zinc has been found effective under similar 
conditions, the topographic features being scratched upon its 
surface and later intensified for transfer by rubbing with lamp¬ 
black and oil. With either of these materials, work can often 
be done on misty days, when the wet from brush and leaves of 
trees would soon soak common paper. 

69. Preparation of Field Sheets.—In planning a plane- 
table survey of a given region a number of plane-table sheets 
should be prepared of such a size as will fit the board. On 
these the work should be so planned as to leave ample 


I 76 SCALES, PLANE-TABLE PA PEP, A N'T PENCILS. 


margin on each edge to permit of transferring from and con¬ 
necting between the various sheets. On each sheet should 
then be platted at least two points (Art. 188), the relative 
positions and distances between and azimuths of which have 
been previously determined by instrumental triangulation of 
primary or secondary order. In case no such prior triangu¬ 
lation exists, a suitable location should be chosen and a tem¬ 
porary base line carefully measured with long steel tapes or 
wire (Art. 204), and plotted on the sheet as nearly as possible 
in correct azimuth and in the relative position which its loca¬ 
tion on the ground bears to the area under survey. From 
the ends a plane-table triangulation may be expanded as 
from located trigonometric positions. In extreme cases, 
where absolute distances are not essential or where work is to 
be checked by an after-primary triangulation, two points 
may be selected as initial stations and their relative positions 
be fixed on the plane-table sheet, the distance being esti¬ 
mated and the azimuth marked by means of a magnetic 
needle. If later a geodetic triangulation locates two con¬ 
nected points and an azimuth within the surveyed area, the 
map may be adjusted. 

Where careful plane-table triangulation is being con¬ 
ducted, points should not be transferred from one plane-table 
sheet to another. Each sheet should have located upon it at 
least two points, the positions of which have been deter¬ 
mined and computed by geodetic methods. If for any 
reason this is impossible, it should be assumed that the act 
of transferring from one sheet to another has distorted or 
affected unfavorably the plane-table triangulation, and in 
order that this shall be in one direction only, and therefore 
susceptible of after-correction, only two points should be so 
transferred, with the intention that ultimately a scheme of 
instrumental triangulation may be extended over the area 
under survey and the plane-table work be adjusted thereto. 

The first desideratum in fastening plane-table *nper to the 


PREPARATION OF FIELD SHEETS. 


177 


board is that it shall be held firmly and equally, and so as 
not to be disturbed in its position by the friction of the 
alidade or by ordinary winds. One means of effecting this is 
by brass spring-clamps; a second is by ordinary thumb-tacks; 
and a third by screw-tacks. The latter are decidedly the 
better. Clamps, being large, are liable to accidental dis¬ 
turbance, they do not hold the paper firmly, and are at all 
times therefore liable to permit a movement of the paper. 
The ordinary thumb-tacks hold the paper firmly when in 
place, but are easily loosened and lost, while in high winds 
the whole paper may be suddenly ripped from the board. 


C 



Fig. 57. —Double Screw to Hold Plane-table Paper. 

A , plane-table board. B , hollow brass wood screw. 

C, milled head brass clamping screw. 

The paper should bear at all times the same relation to the 
board and should be so immovable as to form practically a 
part of it. A thumb-tack which fairly fills the requirements 
has a screw-thread cut on the spike, and the head has holes 
sunk into it so that these may be clamped by a spanner and 
the tacks screwed into the wood. These, however, project 
so as to interfere with the free movement of the alidade. 
The plane-table boards of the Geological Survey have a special 
attachment set into each of the corners and sides, which con¬ 
sists of a brass cylinder having a screw-thread on the outside 
by which it is sunk into and flush with the surface of the 
board, and the inner surface has a female screw, into which a 
milled-head clamping-screw is fastened through the paper. 

(Fig- 57-) 









178 SCALES, PLANE-TABLE PAPER, AND PEN CLLS. 

70. Manipulation of Pencil and Straight-edge.—All 

lines drawn on the plane-table board should be made with the 
hardest of pencils sharpened to a very fine point, and the lines 
should be drawn lightly and carefully and close to the edge of 
the rule. Great care should always be taken to hold the 
pencil in the same position, either very close under the edge 
of the alidade or vertically, so that its point shall be invari¬ 
ably at the same distance from the edge, and the same side of 
the straight-edge should always be employed, lest the two 
sides be not truly parallel or bear a wrong relation to the. 
axis of the telescope. If any part of the straight-edge is 
raised from the paper, especial care must be observed that 
the pencil does not run under its edge and thus deviate from 
the straight line. 

It is desirable not to draw lines the full length of the 
sight, but short lines should be drawn on the paper' approxi¬ 
mately at the location of the point which is sighted, and other 
short lines should be drawn at each end of the straight-edge, 
so that the latter may at any time be laid correctly on the line 
sighted (Fig. 47). The alidade should never be moved by 
sliding it over the surface of the table, but in changing its po¬ 
sition it should be lifted up and carefully set down again on the 
table, so as not to rub the lines or soil the paper. When an 
intersection of two or more lines is obtained, the point located 
should not be pricked with a pin or pencil point, but the 
location should be pricked lightly and finely with a delicately 
pointed needle. A needle-point should never be inserted 
in the paper at the point located so as to be used as the fulcrum 
about which to rotate the alidade, but the latter should al¬ 
ways be lifted up and laid down with its edge against the 

located point and in the same relation thereto as were the 

• 

lines drawn to the point with the pencil; that is to say, the 
under edge of the rule must bisect or be tangent to the point 

according as was the pencil-point in drawing the line which 

— „ , • * * 

produced the location. 


NEEDLE-POINTS\ PENCIL HOLDERS AND SHARPENERS. 179 

* 

71. Needle-points, Pencil Holders and Sharpeners. —In 

running a traverse, and in the execution of plane-table 
triangulation, the little devices and tools with which the 
topographer is provided aid greatly in facilitating his work. 
A fine needle-hole may be made to mark the location of a 
triangulation station. In traversing, however, the work is 
greatly expedited by sticking a very fine needle into the board 
around which to revolve the light sight-alidade. In this 
manner the topographer has not to watch the point on the 
paper to see that his alidade is tangent to it, but has simply 
to press the alidade edge against the needle-point. Such 
needle-points are made by taking a No. 10 needle, breaking it 
in half, and melting a sealing-wax head upon it. In this 
manner the short stem renders it less liable to be broken, and 
the head gives something large enough for the topographer to 
handle readily and press with force into the paper. 

It is slow work attempting to get a sufficiently sharp and 
satisfactory edge on a pencil with a penknife, and as the 
pencil must be sharpened frequently in order to keep it in 
condition for fine work, sand-paper sharpeners, preferably in 
the form of pads, as furnished by dealers, should be provided, 
and these should be carried attached to the board by a string, 
so as to be always at hand for rapid renewal of the pencil- 
point. 

In order that the rubber eraser and the pencil shall be 
always in the most accessible places, leather pencil pockets or 
holders should be provided in which pencils can be carried by 
attaching the holders to the outer garment of the topographer. 
These holders help protect the pencil-point. The rubber eraser 
should either be tied by a string to the board, or, better, metal 
tips provided with rubber should be supplied for all pencils. 
A sufficient number of these should be carried for renewals, 
and thus the rubber is always handy when it is attached to 
the reverse end of the pencil. 


CHAPTER IX. 

PLANE-TABLE TRIANGULATION. 

72. Setting up the Plane-table. —In sighting signals 
these should be bisected as near the base as possible, and 
signal-poles should be straight and perpendicular, and the flags 
upon them white or black according to the color of the back¬ 
ground against which they are to be seen. They should be 
of such size as to be visible at the greatest distance from which 
they must be observed. The positions of the stations should 
be well marked with a small cairn of rocks and by measure¬ 
ment to some near-by witness-mark, so that if the signals are 
disturbed their positions can be readily found. 

The theoretic requirements of setting up a plane-table at 
a station are: 

1. The plane of the board should be horizontal. 

2. The projection of the station on the map should be 
vertically over its position on the ground. 

3. The meridian of the point on the plane-table sheet 
should be in the plane of the meridian of the station. 

The first of these requirements is met by a proper con¬ 
struction of the instrument. For small-scale maps, as those 
- of .more than IOOO feet to the inch, the second requirement 
does not necessitate the plumbing of the platted point exactly 
over the station, since the instrument can generally be set up 
near enough by eye. On maps of larger scales the location 
i&n the plane-table corresponding with the point occupied must 
be plumbed over the latter; that is to say, the center of the 
board is not plumbed over the station-mark, but the platted 
point itself. If the plane-table be set up by eye, it can easily 
be fixed within six inches of its true position. At a range of 

half a mile such an error would subtend an angle of less‘than 

180 


SETTING UP THE PLANE-TABLE. l8l 

a minute, and angular errors of such small amount may easily 
be neglected. 

1 he third of the above requirements is met by orientation 
of the plane-table board. This is its adjustment in azimuth, 
by which all lines joining points on the sheet are made par¬ 
allel to corresponding lines in nature. 

The inclination of the board from the true horizontal 
plane or the amount which it is out of level affects the location 
in azimuth far less than would be at first estimated. This is 
well illustrated in the following table, prepared by Mr. Josiah 
Pierce, Jr. 


Table IV. 

ERROR IN HORIZONTAL ANGLE DUE TO INCLINATION OF 

PLANE-TABLE BOARD. 


Inclination 
of Board. 

Q 

Angle when 
Level. 
a 

Angle when 
Inclined. 

P 

Maximum 

Errors. 

0 

O 

f 

// 

0 

r 

// 

/ 

// 

1 

45 

00 

08 

44 

59 

52 

O 

16 

2 

45 

00 

36 

44 

59 

53 

I 

03 

3 

45 

01 

10 

44 

58 

49 

2 

21 

4 

45 

02 

06 

44 

57 

54 

4 

12 

5 

45 

03 

16 

44 

56 

43 

6 

33 

6 

45 

04 

43 

44 

55 

16 

9 

27 

7 

45 

06 

26 

44 

53 

34 

12 

52 

8 

45 

08 

24 

44 

5i 

33 

16 

5i 

9 

45 

10 

38 

44 

49 

20 

21 

18 

10 

45 

13 

09 

44 

46 

50 

26 

19 

11 

45 

15 

57 

44 

44 

03 

3i 

54 

12 

45 

18 

59 

44 

4i 

01 

37 

58 

13 

45 

22 

12 

44 

37 

42 

44 

36 

14 

45 

25 

55 

44 

34 

05 

5i 

50 

15 

45 

29 

47 

44 

30 

14 

59 

33 


From the above it appears that a plane-table or theodolite 
may be 15 0 out of level before the maximum error in the 
measurement of a horizontal angle will approach i°. 

Also, the error in azimuth is a maximum when (x-f/3=go°. 















PLA NE- TA BLE TP/A NG ULA 7 'ION. 


182 


The above results may be obtained by the following sim¬ 
ple formula: 

0 * 

1 = -- approximately, . . . . (1) 

2 ^ ^ 

in which 6 is the inclination of the board or angle which it 
makes with the horizontal. 

Thus, if the board were out of level i°, the maximum 
error in azimuth which would be produced would be about 
16", an amount scarcely appreciable at 12 feet. An error in 
level of 3 0 would only produce an appreciable maximum 
error of 2' 2 1 ". 

73. Location by Intersection.—There being platted upon 
the plane-table paper (Arts. 69 and 188) the known positions 
of at least two points which are in view from the station over 
which the plane-table is set up, the succeeding plane-table 
triangulation consists in the determination of the relative 
positions on the paper of additional points in nature. This 
should, so far as practicable, be accomplished by the method 
of intersections. This is accomplished by previously occupy¬ 
ing known positions and by constructing a graphic triangula¬ 
tion on the plane-table board from these, including unknown 
positions which are platted in the course of the work. Where 
this is not practicable, as is occasionally the case, because of 
the impossibility of occupying some of the known positions, 
the work must be performed by the method of resections (Art. 
74), by which unknown points are occupied and positions 
determined and platted on the paper by sighting to known 
points. 

The controlling condition in the conduct of plane-table 
triangulation is that the board shall be in orientation (Art. 
72). Let the station P be occupied, and p be its platted posi¬ 
tion on the plane-table board (Fig. 58, A). Let a , b, c be 
the platted positions on the board of the signal A, the church- 
spire By and the flag C. The plane-table board being leveled 



LOCATION BY INTERSECTION . 


■83 




Fig. 58.—Intersection with Plane-tabu*. 
























184 


PLANE- TA BLE TP JANG U LA 7'JON. 


and oriented approximately by eye by ranging the lines from 
p towards a, b, c in the directions of the corresponding signals, 
A, B, C, the edge of the alidade is placed on the line pa, and, 
the horizontal motion being undamped, the board is swung in 
azimuth until the cross-hairs bisect the signal A, when the 
horizontal motion is clamped. The alidade is now placed 
successively on the lines pb and pc. If in sighting the 
signals B and C the cross-hairs bisect these, the instrument is 
oriented. If it does not exactly bisect them, there is some¬ 
thing wrong with the platting on the known points or with 
the observation of one or more of the signals. If the posi¬ 
tions of the points have been accurately determined ana 
platted, the cross-hairs must bisect all of the signals on known 
stations when observed from any known position. 

The orientation of the board being now verified, plane-table 
triangulation is extended by placing the edge of the alidade 
on the point p and swinging it until the cross-hairs bisect a new 
signal, D , towards which a line is drawn. Lines are also drawn 
along the edge of the ruler when it is pointed to a chimney, 
a cupola, or to other visible and easily distinguishable objects, 
the near end of the alidade being of course on the point p „ 
Everything which is observable from this station and which 
may possibly be recognized from succeeding stations being now 
indicated on the paper by lines drawn from p, the alidade may 
be moved and sighted successively to each of the points ob¬ 
served, and the vertical angle read to them and recorded 
(Art. 160). The work of this station is then completed, and 
the topographer moves to the next station, A. 

Having oriented the board on the second station as before, 
by placing the edge of the alidade against the known and 
occupied point a and sighting successively to the known points 
P, B , C, etc., the orientation is verified by observing if the 
edge of the alidade passes through the located points p, b, c, 
etc. If so the topographer proceeds to intersect some of the 
lines previously drawn from the first station, P. The line 


LOCATION BY RESECTION. 


I8 5 


% 0 

drawn towards the new point, D, intersects the line drawn 
from P in the point d , which is its position (Fig. 58, B). 
Likewise intersections are made on the cupola, the chimney, 
etc., by sighting these and drawing lines along the edge of 
the ruler. 

The positions of the points thus determined are not con¬ 
sidered sufficiently well established for the propagation of 
triangulation unless a third intersection is had on them foi 
the purpose of verification. Where it is difficult to get a third 
intersection, locations by two lines will answer sufficiently 
well for intermediate or tertiary points, but every effort 
should be made to get a third intersection (Fig. 58, C), pro¬ 
viding anything of moment is dependent upon the position. 
The third intersection is had as in the case of the previous 
ones by occupation of one of the remaining points, B or C, or 
perhaps by occupation of the new point, D. In the latter 
event, only two lines having been previously drawn through d, 
its position is more accurately verified after orientation on the 
previously occupied stations P and A by resection from the 
occupied stations B and C. In this event it may be necessary 
to unclamp the board and swing it a trifle in azimuth as 
described in the three-point problem (Art. 75), in order to get 
a more exact location than is given by two intersections. 

74. Location by Resection.—The three-point problem calls 
for the finding of distances from an unknown and occupied 
point to three others whose relative positions and distances 
are known. Only the constructive or graphic solutions of the 
problem are here given, and not the theoretic or trigonometric, 
since the operation of locating a point on the plane-table is 
graphic and not trigonometric. 

The determination of an unknown point graphically on 
the plane-table is performed by the method of resection, 
which consists in the occupation of the unknown point with 
the plane-table and the sighting from it to the three known 
points, on which well-defined signals must be erected and the 


PL A AE- TA BLE 7 'ElA AG U LA TION. 


186 

positions of which are previously plotted upon the plane- 
table sheet. The determination of the unknown position 
may be accomplished by several methods, the earlier of which 
is known as Bessel’s, from its inventor, though the most sat- 
isfactory method, and that now almost universally employed, 
is known as the Hergesheimer or Coast Survey method. In 
addition there is an approximate but rapid and practical 
method by means of tracing-paper, generally known as the 
graphic method, and there are also the less well-known and 
rarely employed Lehmann s and Netto’s methods. 

75. Three-point Problem Graphically Solved_Three 

simple, practical rules for determining the location of an un¬ 



known point on the plane-table by means of the three-point 
problem are the following (Fig. 59): 

1. When the new point is on or near the circle passing 
through the other points, the location is uncertain. 











THREE-POINT PROBLEM. 


<3 7 


2. When the new point is within the triangle formed by 
the three points, the point sought is within the triangle of 
error. 

3. When the new point is without the circle, orient on 
the most distant point, then the point sought is always on the 
same side of the line from the most distant point as the point 
of intersection of the other two lines. 

The last rule is that most usually called into requisition, 
and is perhaps the most important in aiding in the quick de¬ 
termination of the unknown position. 

76. Tracing-paper Solution of the Three-point Prob¬ 
lem.—By the use of tracing-paper the three-point problem 
is solved approximately with great rapidity. Setting-up the 
table on the unknown point /’(Fig. 58), fasten on it a piece 
of tracing-paper of sufficient size to include the positions 
of all four points. A fine point is marked upon it to repre¬ 
sent the position of p and as near the actual location of that 
point on the paper can be estimated by eye. The alidade is 
then centered about the point p and pointed successively at 
the three known points A, B } and C, and the lines pa , //;, 
and pc are drawn on the tracing-paper. The alidade being 
then removed and the tracing-paper released, this is so shifted 
over the plane-table sheet that the line pa shall always pass 
through the located point a , the line pb through the located 
point b } and the line pc through the located point c. Then, 
with all three lines passing through the known points, the 
point / is exactly over its correct position on the plane-table 
paper, and may be pricked through to the latter. 

As this method is approximate only because of the little 
inaccuracies introduced in stretching the tracing-paper; or 
because of its wrinkling and the difficulty of drawing very fine 
lines on the tracing-paper and properly superimposing this, 
it is well, where an exact location of d is desired, to then test 
the position of the latter by resection from the known points, 
when a small triangle of error may be found. This will be so 


188 


PLA NE - TA BLE TP I A NG ULA TION . 


small, however, that a trifling movement of the board will 
bring the table into exact orientation, and frequently with 
much greater accuracy and ease than by using the graphic 
three-point method only. 

77. Bessel s Solution of the Three-point Problem.— 

Bessel had two methods of solving this problem, only the first 
of which will be described, as the other is less practical. The 
plane-table is put in position at the unknown station from 
which the three known points must be visible, and the position 
of the unknown point can then be found as follows, provid¬ 
ing it be not in the circumference of a circle passing through 
the three fixed points: 

A quadrilateral is constructed with all the angles within 
the circumference of a circle, one diagonal of which passes 
through the middle one of the three fixed points and the point 
sought. On this line the alidade is set, the telescope directed 
to the middle point, and the plane-table oriented. Resections 
upon the extreme points intersect on this line and determine 
the position of the point sought. In Fig. 60, let a , b , c be 
the platted position of the known points; the plane-table 
being set up on the unknown station D and leveled, the alidade 
is set on the line ca, and the end at a is directed, by revolving 
the table, to its corresponding signal A, and the table clamped ; 
then, with the alidade centered on c, the middle point B is 
sighted with the alidade and the line ce drawn along the edge 
of the rule; the alidade is then set upon the line ac, and the 
telescope directed to the signal C by revolving the table, and 
the table clamped. Then, with the alidade centering on a , the 
telescope is directed to the middle signal B, and the line ae is 
drawn along the edge of the rule. The point e (the intersec¬ 
tion of these two lines) will be in the line passing through the 
middle point and the point sought. Set the alidade upon the 
line be, direct b to the signal B by revolving the table, and the 
table will be in position. Clamp the table, center the alidade 
upon a, direct the telescope to the signal A, and draw along 


THREE-POINT PROBLEM . 1 89 

0 

the rule the line ad. This will intersect the line be at the 
point sought. Resection upon C, by centering the alidade on 
c in the same manner as upon A, will verify its position. 



Fig. 60.—Bessel’s Graphic Solution of the Three-point Problem. 


In the use of the Bessel methods for the determination of 
position, the triangle formed by the three fixed points can be 
contracted or extended as may be desirable, by drawing a line 
parallel to the one joining the two extreme points, terminated 






















190 


PLA NE- TA BLE TRIA NG ULA TI'ON. 


by those joining the extremes with the middle point. The 
lines laid off at these representative extreme points, in the 
manner described for the extremes, will intersect in the line 
passing through the middle point and the point sought. 

This affords the means of using a point in view which 
would not be within the size of the table when the other two 
points are shown, by contracting the triangle formed by the 
three points until both extremes are brought within the table 
size and within reach of the alidade. A resecting line for the 
point off the table can be drawn from its signal near the esti¬ 
mated position of the point sought, and a line drawn through 
the corresponding point off the table, parallel to this, will 
determine the precise position of the point sought, to be veri¬ 
fied by resection on the other extreme point. 

78. Coast Survey Solution of the Three-point Problem- 
—This method depends upon the fact that when the plane- 
table is set up and is not in orientation, resection from any 
three known points, except from a point on the circumference 
of a circle passing through these points, will form a triangle 
called the triangle of error, or two of these lines will be par¬ 
allel and intersected by the third. The position of the true 
point can then be determined graphically from these several 
intersections, and is always at the point of intersection of the 
arcs of the circles drawn through each two points and the 
point of intersection of the lines drawn from them. There 
are numerous practicable modes of locating the point sought, 
and these have been divided into several classes, and these 
again into several cases or subdivisions for convenience of 
description (Fig. 59). This classification is based upon the 
location of the true point in relation to the triangle of error, 
the triangle formed by the three fixed points being called the 
great triangle, and the circle passing through these points the 
great circle. The topographer is supposed to face the signals, 
and directions right and left are given accordingly. 

Class i. When the point sought falls within the great 


RANGING-IN AND LINING-IN. 191 

0 

triangle, the true point is within the triangle of error. If the 
line from any of the view-points falls to the right of the inter¬ 
section of the other two points, turn the table to the left; 
and if to the left, turn it to the right. 

When the point sought is without the great triangle, the 
true point is also without the triangle of error and is situated 
to the right or left of it, according as the table is out of posi¬ 
tion to the right or left. 

Class 2. When the point sought falls within either of three 
segments formed between the great circle and the sides of the 
great triangle, the true point is on the side of the line from 
the middle point opposite to the intersection of the lines from 
other points. Also, where the three fixed points are in a 
straight line, in which case the three points are considered as 
being on the circumference of a circle of infinite diameter, the 
true point always lies within one of the segments of the great 
circle. 

0r 

If the line from the middle point is to the right of the 
intersection of the other two, turn the table to the right, and 
if to the left, turn it to the left. 

Class 3. When the point sought falls without the great 
circle and within the sector of either angle of the great tri¬ 
angle, the true point is on the same side of the line from the 
middle point as the intersection of the lines from the other 
two points. 

If the line from the middle point is to the right of the in¬ 
tersection of the other two, turn the table to the left, and if 
to the left, turn it to the right. 

Class 4. When the point sought is without the great circle 
and the middle point is on the near side of the line joining 
the other two points, the true point is without the triangle of 
error, and the line drawn from the middle point lies between 
the true point and the intersection of the other two lines. 
Also, when the point sought is on the range of either of the 
two points, and the table deflected from the true position, the 


192 


PLANE- TABLE TRIANG ULA TION. 


lines drawn from these points will not intersect, but will be 
parallel to and intersected by the line drawn from the third. 
The true point is then between the two parallel lines. 

When the line from the right-hand station is uppermost, 
turn the table to the right, and when that from the left is 
uppermost, turn the table to the left. 

79. Ranging-in, Lining-in, and Two-point Problem.— 
It is sometimes desirable to place the plane-table in position 
at an unknown point from which only two known points are 
visible. This may be easily done in the following two cases 
by methods known as “ ranging-in ” and “ lining-in.” 

Ranging-in consists in determining the position of a point 
on a line already drawn on the plane-table, but elsewhere on 
that line than at the position of the point sighted* In Fig. 
61 let A and B be the positions of the two known points, and 
let AC be a line drawn from A towards the point C. When 



Fig. 61.—Ranging-in. 


the topographer reaches C let him find it, for some reason, in¬ 
accessible; it may be a tree, a building, or some other object 
near which, but not over which, he may set up. Aligning 
himself, therefore, by eye in the direction AC by means of 
range-poles or by sighting over the top of C at A, he sets 
up the plane-table on the line thus sighted by placing the 
alidade on the line ac and resecting on A, and clamping the 
table, when it will be in orientation. Placing the end of the 
alidade now on the point b and resecting on B, the line drawn 
along the edge of the rule will intersect the line ac at the 
point c\ the position sought. 

In lining-in , the positions of the points A and A (Fig. 62) 
are known and located on the plane-table sheet at a and b, and a 



TWO-POINT PROBLEM. 


193 


line having been drawn from one of the stations A towards an 
nndeteimined point, 0 , it is desired to locate another undeter¬ 



mined point, C' t on the line ac, but at a considerable distance 
from either point. The topographer, finding a suitable posi¬ 
tion near C r , proceeds with the aid of an assistant, d, to place 
himself in a line between A and C. Standing some distance 
apart, they line one another in, the topographer, c', by sighting 
over his assistant, d, at A; the assistant, d, sighting over the 
topographer, d , at C ; then they motion each other backwards 
and forwards at right angles to the line ac until each finds the 
other exactly in line with his range-point. The topographer 
is then on the line sighted from A to C, and may set up his 
plane-table and, placing the alidade on ac, resect on A, when 
the board will be in orientation. Now, setting the alidade on 
point b and resecting on B , a line drawn along the edge of 
the rule will intersect the line ac in the undetermined points'. 

A more difficult case of making a location by the two-point 
problem is the following: Two points A and B (Fig. 63), not 
conveniently accessible, being located on the paper at a and b, 
it is desired to put the plane-table in position at a third point, 
C. A fourth point, D, is selected, such that the intersection 
from C and D upon A and B make sufficiently large angles for 
good determinations. Put the table approximately in position 
at D, by estimation or compass, and draw the lines Aa , Bb , 
intersecting in d\ through d draw a line directed to C. Then 
move to and set up at C, and assuming the point c on the 
line dC , at an estimated distance from d, and putting the 








194 PLANE- TABLE TRIA NG ULA T 1 ON. 

table in a position parallel to that which is occupied at D, by 
means of the line cd draw the lines from c to A, and from c to 

B. These will intersect the lines dA, dB at points a' and b' y 

♦ 



which form with c and d a quadrilateral similar to the true 
one, but erroneous in size and position. 

The angles which the lines ab and a b' make with each 
other is the error in position. By constructing now through 
c a line cd' making the same angle with cd as that which ab 
makes with a'b\ and directing this line cd' to D, the table will 
be brought into position, and the true point, c , can be found by 
the intersections of aA and bB. Instead of transferring the 
angle of error by construction, it may be convenient to pro¬ 
ceed as follows, observing that the angle which the line ab' 
makes with ab is the error in the position of the table. As 
the table now stands, ab' is parallel with AB, but it is desired 
to turn it so that ab shall be parallel to the same. If, there¬ 
fore, the alidade be placed on a'b' and a mark set up in that 
direction, then placing the alidade on ab and turning the table 
until it again points to the mark, ab will be parallel to AB 
and the table be in position. 












CHAPTER X. 


TRAVERSE INSTRUMENTS AND METHODS. 

8o. Traverse Surveys. —In making topographic surveys— 

1. The area mapped may of necessity be surveyed by 
running meander or traverse lines where it is impossible or 
impracticable to conduct the work by. triangulation; or 

2. Traverse lines may be run in conjunction with a 
trigonometric survey to fill in the details which cannot be 
economically reached by such methods. 

Rarely can a topographic survey be made in the most 
satisfactory manner by trigonometric methods alone and 
without the aid of traverse work. Such conditions may be 
met in country of bold features, quite open, where numerous 
natural objects may be at all times visible for triangulation 
intersection or where stadia-rods or flags may be readily seen 
from every station occupied. Ordinarily, in any country the 
lower lines of the terrane are not visible from the triangula¬ 
tion stations, and therefore their topography is most easily 
obtained by means of traversing. 

In running traverse surveys the errors naturally due to the 
measurement of distances and azimuths are of such amount as 
to be perceptible in maps of almost any scale, and they must 
therefore be adjusted or eliminated by tying either to 
traverses of greater refinement (Arts. 82, 226, 345a) or to posi¬ 
tions located by the trigonometric survey (Art. 73). Traverses 
made in connection with topographic mapping are of several 
degrees of accuracy, according to the amount of trigonometric 
or other control available for their adjustment. Where the 

*95 


ig6 TRA VERSE INSTRUMENTS AND METHODS. 


summits are of comparatively uniform elevation and are 
timbered, and it is therefore difficult to conduct triangula¬ 
tion, it may be more economical to control the surveys 
by traverses. In making surveys in this manner, covering 
large areas on small scales, as i or 2 miles to the inch, 
primary traverses (Art. 226) are run about the area to be 
surveyed, and these are executed with the greatest care, 
almost as in the measurement of base lines (Art. 202), and 
they are adjusted to one or more astronomic positions (Part 
VI). Between such primary traverses the topographer will 
run secondary traverses with transit (Art. 85) or plane-table 
(Art. 81), or solar alidads (Art. 345a); distances being meas¬ 
ured by stadia, chain, or odometer, according to circumstances. 

Where the roads are level, have few short bends, are mostly 

% 

in long tangents and are open, measurements may be made 
with nearly as great accuracy by means of the odometer 
(Art. 98) as by stadia or chain. Where the roads are crooked 
or it is necessary to run traverses off them and across 
country, stadia measurement (Art. 102) should have the 
preference, providing the timber is not so dense as to preclude 
its use. In densely wooded country the chain or tape (Arts. 
97 and 99) must be employed to measure distances. When 
it becomes necessary to procure additional elevations in con¬ 
junction with the traverse, the stadia is most economical, 
since vertical angulation may be carried on at the same 
time. 

Where traverses are run in connection with small-scale 
geographic mapping (Art. 29), merely to get the directions 
and bends in roads and trails, the crudest methods are per¬ 
missible, because of the numerous points on these which will 
be sketched-in from the plane-table stations. Under such 
circumstances the prismatic compass (Art. 91) and measure¬ 
ment of distance by odometer, by pacing, or by counting 
the paces of animals (Art. 95) with notes, kept in a book, 
will furnish sufficiently ample results. Where the command 


TRAVERSING BY PLANE-TABLE AND COMPASS. 197 

of the terrane from the plane-table stations is incomplete, and 
traverses must be run either to obtain the positions and 
directions of roads or to map adjacent topographic data, 
traverses should be run with light plane-table and sight- 
alidade (Arts. 61 and 62), accompanied by distances measured 
with odometer, stadia, chain, or pacing. Where traversing 
is done not only to get roads and topographic detail, but also 
to furnish secondary and tertiary control, plane-tables of the 
Johnson pattern (Arts. 58 and 59) should be employed, 
with telescopic alidade or with solar alidade (Art. 34'tf). 
Along the line of the traverse the sight-alidade Art. 62) 
should be used on most of the intermediate locations, 
and distances should be measured as in tha previous 
case. 

81. Traversing by Plane-table and Magnetic Needle. 

—In all traverses for small-scale maps, the plane-table can 
be most satisfactorily oriented by means of a compass-needle. 
In work of this character a substantial plane-table is not 
necessary, a light portable one being most satisfactory. This 
may be either of the Johnson form (Art. 58), where a tele¬ 
scopic alidade is to be used in order that vertical angles and 
stadia measurements may be taken (Arts. 160 and 102); or if 
a sight-alidade will suffice for the work to be performed, 
the traverse-table should be of the simplest form possible 
(Art. 61). 

Traverses run with this apparatus in conjunction with 
odometer or stadia measurements (Arts. 98 and 102) will 
usually close in short circuits of ten to thirty miles perimeter, 
with errors so small as to be readily adjusted by connection 
to better traverse or triangulation locations. In conducting 
traverses by this method back-flags are unnecessary, and fore¬ 
flags are only necessary in large-scale work (Chap. III). 
Work on scales smaller than one mile to the inch and where 
the traversing is on roads requires no fore-flags, as the direc¬ 
tion of the road itself affords sufficient guide to the direction of 


I9§ TRAVERSE INSTRUMENTS AND METHODS. 

the sight taken. Where traversing is across country and 
without guide of road or stream line, some signal, as a rod or 
man, is necessary to serve as a foresight and mark the fore¬ 
station. In traversing in this way it is unnecessary to set up 
the instrument at every station, for, as the orienting is done 
by the needle, it can be done with greater satisfaction by 
the occupation of every alternate station only, whereas the 
speed of thus setting up only at alternate stations is greatly 
increased. 

Set up the plane-table at the first station, A , and orient 
(Art. 72) by swinging the board into such position that the 
needle will point to the north and south marks. The 



Fig. 64. —Traversing with Plane-table. 

foresight is taken by placing the near end of the alidade 
against the known point, a , and sighting in the direction of 
the road or to a fore-flag, B (Fig. 64), a short pencil-line 
being drawn along the edge of the ruler. Moving forward 
now to the foresight point B , the traverseman notes the dis¬ 
tance and continues on to the next bend in the road or 







CONTROL BY I.ARGE-SCALE MAGNETIC TRAVERSE. 1 99 

* 

traverse line, where he sets up his instrument at C, again 
orients by the compass-needle, and at once plats on the first 
foresight the distance to the first fore-station b. Now plac¬ 
ing the far end of the alidade against the second location b r 
and revolving the alidade about it, he backsights on the 
station B, and draws a line towards his present station, from 
which he measures off the distance from the second to the 
third or present station, the position of which is then deter¬ 
mined and platted at c. The result is to give him on his 
board two lines and three points in the traverse. He then 
proceeds as before, by observing a foresight on the next for¬ 
ward station, D, and moving on beyond it to the next sta¬ 
tion, E. 

82. Control by Large-scale Magnetic Traverse with 
Plane-table. —When it is necessary to secure secondary 
control quickly and for but limited areas, this may be graphi¬ 
cally done on the plane-table more conveniently than by 
using a transit and computing latitudes and departures (Art. 
90). The process is by traversing with compass-needle for 
scales less than 1 : 10,000, as above described, but the scale 
of platting must be increased to two or three times that 
chosen for the field map. This is in order to eliminate the 
errors in measurement of distances, and also those due to the 
graphic platting of the azimuths. Such errors as occur in 
running the traverse will be largely eliminated by its reduc¬ 
tion to the smaller scale on which the remaining field-work 
is to be done. As described in Art. 69, the location of the 
initial point may be platted on the plane-table sheet, or, if 
not known, may be assumed, in which case the work will be 
started in such position on the board as will permit of the 
greater extent of the traverse coming within its area. 

In this way a number of traverse lines on the larger scale 
are run back and forth across the board from one known point 
to a terminus at another known point, perhaps thirty or fifty 
miles distant. The work must be performed on a large 


200 TRAVERSE INSTRUMENTS AND METHODS. 

plane-table board with a telescopic alidade (Arts. 57 and 59). 
Distances must be measured with a stadia or chain (Arts. 102 
and 99), and the former or a flag must be sighted to give 
directions more accurately. A long azimuth line parallel to 
the compass-needle is drawn the full length of the sheet. 
The compass should be a declinatoire of about 5 degrees 
range, the needle being not less than 6 inches in length, and 
this should be set in a brass box let into the board and 
parallel to one of its sides (Art. 61). 

On completion of the traverse, a projection is made (Art. 
184) on the same scale, and on it are platted the initial and 
closing known points (Art. 188). The traverse is then 
transferred to this projection by means of the long orienting 
lines, and if run with care will close between the two known 
points within a reasonably small limit of error, perhaps a 
tenth of an inch. Controlling points on this traverse, as road- 
crossings, buildings, etc., are then transferred by propor¬ 
tional dividers or by measurement of their positions with 
relation to projection lines to another projection which is 
platted on the scale of the topographic field-work, probably 
two or three times smaller than that of the traverse. This 
reduction will diminish the closure error to such an extent 
that on the topographic field scale it may be a twentieth of 
an inch or less. This will probably be sufficiently close to 
serve all the purposes of secondary control on which to tic 
additional traverses. These may now be run with less ac¬ 
curacy (Art. 81), as they are only to obtain details of topog¬ 
raphy (Arts. 12 and 16). 

83. Traversing by Plane-table with Deflection Angles. 
—Where plane-table work is being executed on a scale larger 
than 1000 feet to 1 inch, directions should be by deflections 
from back-flags. Where, however, traversing is platted to 
smaller scales than, say, 1 : 10,000, they can be executed 
with greater precision by means of a plane-table oriented by 
magnetic needle. 


PLANE-TABLE TRAVERSE BY DEFLECTIONS. 


201 


In traversing with plane-table and deflection angles on a 
large scale, the plane-tabler will set up at the first traverse 
station. If this is located on his map by intersection from 
triangulation, or is a point on a line the azimuth of which is 
known, he is at once prepared to proceed with his traverse. 
The position of his station may not be known on paper, in 
which case it may be obtained by resection from three visible 
platted points (Art. 74), or he may have no way of fixing the 
position on the plane-table. Making a fine mark on the 
paper by means of a sharp-pointed needle, and accepting this 
as the position of his station, he proceeds with the traverse in 
the anticipation that the latter may ultimately connect with 
some known point, thus furnishing data from which to make 
adjustment. 

Setting up the plane-table at the first station, A (Fig. 64), 
and accepting its known or assumed location, a , on the board,, 
the traverseman proceeds by orienting (Art. 72) on some 
known point or azimuth line, iT, if he has such, or by placing 
his plane-table as nearly as possible in magnetic meridian by 
needle or by eye estimate. He then rotates the near end of 
his alidade about the occupied point a , and sights over its far 
end at a stadia-rod or other flag, B , for the first foresight. 
Moving ahead now to the new position, he leaves either a 
rodman or a stake or sapling with a piece of cloth or paper 
on it as a back-flag at A. Setting up now at this first fore¬ 
sight station, B , and carefully plumbing over it, he orients by 
placing the alidade on the line just drawn and sights back 
with the alidade to the rear flag A by revolving the table, the 
undetermined end of the line on the plane-table sheet being 
towards him. Knowing the distance from the first station to 
the point B now occupied, either by stadia chain (Arts. 102 
and 99), or other measure, he plots this to scale on the line 
first sighted, and the resulting point is the new position, b , on 
the plane-table. The traverseman, now having his present 
station located and the table oriented in relation to the first 


202 


TRAVERSE INSTRUMENTS AND METHODS. 


foresight, revolves the alidade about the present point, b y 
sights the next fore-flag, C, and draws a line along the edge 
of the ruler. He now moves to the next station, C , and pro¬ 
ceeds as before. 

84. Intersection from Traverse. —In running traverses 
to obtain minor control and to furnish details of topography, 
it is necessary that the traverseman locate by intersection as 
many of the near-by features as practicable, that these may 
act as guides for the control of the sketching and aid in the 
determination of additional elevations. These intersections 
are also essential as aids in the adjustment of traverses 
(Art. 12), for some of the neighboring summits and prominent 
objects located from the traverse will also be located by the 
plane-table triangulation (Art. 73), and they thus furnish a 
means of adjusting the traverse to the triangulation. 

The mode of obtaining these intersections is as follows: 
The traverseman having set up and oriented his plane-table 
(Art. 81), either by backsight or by compass-needle, accord¬ 
ing to the mode of traversing, and having completed the 
observing and platting of the necessary fore and back 
sights for the continuation of his traverse, he places the 
needle at the occupied station, a (Fig. 64), and swinging the 
alidade about this, sights consecutively to such prominent 
objects, v, w , ur, and y, as may be in view and may possibly 
be seen from some of the succeeding traverse stations. To 
each of these he rules a short, li ght 1 ine. Moving on now to 
the succeeding stations, B , C, and D , as any of the points 
previously sighted or additional useful points come into view, 
as at D t radial lines are drawn to them from d y and the inter¬ 
sections of these with the lines from a gives the positions 
of the points v , w, etc. The location of some of these 
•points having been fixed by one or more intersections from 
•the traverse, their elevations may be determined by the 
vertical angle read to them with the telescopic alidade or 
the vertical-angle sight-alidade (Arts. 59 and 62); the angle 


ENGINEERS' TRANSIT. 


203 


read with the distance which can be measured from the plane- 
table furnishing data from which to compute, or obtain from 
tables, differences of elevation (Art. 163). 

85. Engineers’ Transit. —This is the instrument com¬ 
monly employed by surveyors for the angular measurement 
of directions. It consists of a telescope supported in axes, 
called wyes, in which it can revolve in a vertical plane while 
the whole revolves in a horizontal plane, the amount of both 
movements being measured on graduated circles read with 
verniers. There are suitable attachments for clamping the 
telescope and the horizontal circle, and for moving them 
slowly by means of an apparatus called a tangent screw. 
Finally the whole may be revolved about a second horizontal 
axis (Fig. 65). The transit is an instrument but little used 



by topographic surveyors, and is so commonly employed in 
ordinary surveying and described in text-books and catalogues 
that its description will not be elaborated here. There are 
various forms, sizes, and patterns of transits, differing with the 
ideas of the makers and the work for which they are intended, 
and the catalogues furnished by the makers thoroughly 
describe the modes of adjusting and using these instruments. 






























































































204 TRAVERSE INSTRUMENTS AND METHODS. 

The chief points to be remembered in selecting a transit 
are the work for which it is to be used. If the best work is 
not to be executed, and portability is an object, a light 
mountain transit with circles reading to but one minute will 
be sufficiently accurate. If the highest grade of work is to 
be performed, large, heavy instruments having circles read¬ 
ing to twenty or thirty seconds, shifting centers and large 
bubbles, should be employed. If the instrument is to be 
used for trigonometric work, the most important points, aside 
from the graduation of the horizontal limb or circle, are the 
size of the objective of the telescope and its magnifying 
power. That sights may be observed in hazy weather, the 
objective should admit the greatest possible amount of light, 
and it should therefore have but two glasses and be inverting. 

86 . Adjustments of the Transit.—The transit is em¬ 
ployed primarily for measuring horizontal angles between 
two objects which may not be at the same elevation. There¬ 
fore, after pointing at one of these, the telescope has to be 
moved through a vertical arc to bring the line of sight from 
the first point to the second. Any error in the instrument 
which throws the line of sight or line of collimation of the 
telescope out of plumb in performing this operation will 
affect the horizontal angle read. It is therefore evident that 
this adjustment, known as the collimation adjustment, which 
makes the telescope revolve in a true vertical plane, is one of 
the most important. Likewise the vertical axis of the transit 
must be truly vertical in order that when the instrument is 
turned in azimuth the line of sight projected into the horizon¬ 
tal plane may move horizontally. 

The various adjustments of the transit consist each of 
two operations: (i) the test to determine the error, and 
(2) the correction of the error found. If the transit were in 
perfect adjustment— 

1. The object-glass and eyeglasses would be perpendic¬ 
ular to the optical axis of the telescope at all distances; 


AD JUS TMENTS OF THE TRANSIT. 20 5 

2. The line of collimation would coincide with the optical 
axis, and 

3. It would be parallel with the telescope-level, and 

4. It would pass through and be perpendicular to the 
horizontal axis of revolution. 

These salient facts should be ascertained to assure the 
perfect adjustment of the transit. 

The first adjustment is that of the level-bubbles. After 
setting up the instrument make the two small levels each 
parallel to a line joining two opposite leveling-screws; then, 
by turning the leveling-screws so that both thumbs move 
inwards or outwards, bring the bubbles to the center of the 
tubes. 

Turn the instrument 180 degrees in azimuth, and if the 
bubbles still remain centered, the levels are in adjustment. 
If they do not remain in the centers of their tubes, bring 
them back half-way by means of the leveling-screws, and the 
remaining half-way by means of the adjusting-screws at the 
end of each leveling-tube. Repeat the operation several 
times, until the bubbles remain in the centers of their tubes 
when the instrument is revolved. 

To make the vertical cross-hair perpendicular to the plane 
of the horizontal axis , focus the cross-hairs by the apparatus 
at the eye end of the telescope ; then adjust the objective upon 
some well-defined object at a distance of a few hundred feet. 
Move the horizontal limb so as to bring the vertical wire 
against the edge of a building or of a plumb-line or other 
vertical object. Clamp the instrument and note if the 
vertical wire is everywhere parallel to the vertical line. If 
not, loosen the cross-wire capstan-screws and, by lightly 
tapping their heads, move the cross-wire ring around until the 
error is corrected. 

To adjust the line of collimation, which brings the inter¬ 
section of the wires into the optical axis of the telescope, 
point the instrument at some well-defined object at a distance 


206 traverse instruments and methods. 

of several hundred feet and, having made the previous adjust¬ 
ments, clamp the lower horizontal motion and revolve the 
telescope completely over, so as to point in the other direc¬ 
tion. Place there some well-defined object, as a tack in the 
end of a stake, and at practically the same distance from the 
instrument as the first object selected. Unclamp the upper 
plate and turn the instrument half-way round or through 
180 degrees, as indicated by the vernier, and direct the tele¬ 
scope to the first object sighted, B (Fig. 66). Again bisect 


D 



Fig. 66.—Collimation Adjustment. 

this with the wires, clamp the instrument, and revolve the 
telescope over and observe if the vertical wire bisects the 
second object, C , when the telescope is now pointed at it 
from the reverse position. If it does, the line of collimation 
is in adjustment. If not, the second point observed, as E , 
will be double the deviation of that point from the true 
straight line, as the error is the result of two observations 
made when the wires were not in the optical axis of the tele¬ 
scope. In the last pointing of the instrument, after the 
telescope was directed the second time to B> the point 
bisected at E was situated as far to one side of the true 
straight line, BAC , as was the point first sighted, D , on the 
other side. To correct the error, use the capstan-head screws 
on the side of the telescope and move the vertical wire to one 
side or the other by one-fourth the distance, keeping in mind 
the fact that the eyepiece inverts the position of the wires, and 
that in moving these screws the observer must operate them as 
if to increase the error noted. Unclamping the instrument and 
swinging it around so as once more to bisect B, again revolve 
the telescope, and if the adjustment has been correctly made 





TRAVERSING WITH TRANSIT. 


207 


the wires will now bisect the central point, C. Test the 
adjustment by revolving the instrument half-way round again, 
fixing the telescope on B , clamping the spindle, and once 
more revolving the telescope on C> and repeat the observa¬ 
tions and adjustment of the wires until the correction of the 
collimation is completed. 

The adjustment of the standards is the next and last im¬ 
portant adjustment of the transit, and this is made in order 
that the point of intersection of the wires shall trace a vertical 
line as the telescope is moved up and down. This result is only 
obtained when the two standards which support the axis of the 
telescope are at the same height. Point the telescope to some 
object which will give a long vertical range, as at a star and 
its reflection in a bath of mercury, or the top of a tall church 
spire and the center of its base, or a long plumb-line. Fix 
the wires on the top of the object and clamp the spindle, 
then bring the telescope down until the wires bisect some 
good, well-defined point at the base. Turn the instrument 
half-way round or through 180 degrees, revolve the telescope, 
and focus the wires in the lower point. Clamp the spindle 
and raise the telescope again to the highest point. If the 
cross-hairs again bisect it, the adjustment is perfect; if they 
pass to one side, the standard opposite to that side is 
highest, the apparent error being double. This is corrected 
by turning a screw underneath one of the axes which is made 
movable, the correction being made for half of the amount 
of the apparent error. 

87. Traversing with Transit. —A traverse line executed 
with the transit differs from one executed with the plane- 
table or the theodolite because of the ability to transit the 
telescope or revolve it through 180 degrees vertically. As a 
result of this construction of the instrument the angle 
between backsight and foresight which is read and recorded 
is not the full horizontal angle observed by swinging the 
instrument in azimuth, but it is the deflection of the new 


208 


TRAVERSE INSTRUMENTS AND METHODS. 


direction, or of the foresight, to the right or left of the back¬ 
sight prolonged. 

Having set up the instrument at A (Fig. 67), direct the 
telescope at the first point in the traverse B , with the grad¬ 
uated circle set at zero and by using the lower motion. 



Fig. 67. —Traversing with Transit. 

Record the angle zero and the distance AB measured by 
chain, stadia, or other method. Move to B, and setting up 
and plumbing the instrument at that point, backsight on the 
point A , using the lower motion and with the circle still at 
zero. Clamp the lower motion and transit the telescope. 
The instrument will now point in the direction of A', which 
is the prolongation of AB if the collimation be in perfect 
adjustment. Loosen the upper clamp and point at the new 
foresight C , and then reclamp the vernier. The angle a 
is the deflection from the straight line A A' to the right 
towards C. In like manner the instrument is moved to C , and 
the line BC prolonged to B ' by transiting the telescope, and 
the angle ft, from B f to D, is recorded as a left deflection. 


EXAMPLE OF TRANSIT NOTES. 


Sta. 

Distance. 

Deflections. 

True 

Bearing. 

(Azimuth.) 

Mag. Bearing. 

Remarks. 

78 + 80 

Feet. 

765 

O 

CO 

w 

1 

187° 09' 

N. 4 0 30' E. 

Road crossing. 

71 + 15 

Il8o 

-f- 27 0 06' 

205° 51' 

N. 22° 30' E. 

House. 

59 + 35 

435 

+ 6° 27 ' 

178° 45 ' 

N. 5 0 00' W. 

Stream to right. 

55 



172 0 18' 

N. io° 30' W. 

House. 


The notes of such a transit traverse are kept in the follow¬ 
ing manner: In the first or “ Station ” column is recorded 




















TRAVERSING WITH TRANSIT. 20Q 

the total distance in hundredths plus single feet from the 
initial point. In the second or “Distance” column is 
recorded the distance between two stations A and B. In the 
third column is recorded the deflection angle with plus or 
minus signs, according as the deflection is to right or left. 
In running a simple traverse nothing further is requisite than 
the above. If, however, as is likely to be the case in topo¬ 
graphic surveying, it is desired to know also the bearing of 
the line, the true azimuth of some sight, preferably the first 
line, should be determined by observation on Polaris (Art. 
312), and the magnetic declination (Art. 92) should be noted, 
as well as the true or transit declination, by reading the angle 
between the azimuth line and the first line of the traverse. 
This angle should be recorded in the fourth column, “ True 
Bearing.” Then, as the traverse is run, the deflection right 
or left should be added to or subtracted from the last true 
bearing and thus give the new true bearing. For a check 
the compass-needle should also be read and recorded in the 
column “ Magnetic Bearing,” and the true bearing should 
agree with this approximately by the difference of the 
declination observed. There should be a last column in 
which to record remarks of streams passed, road junctions, 
etc. 

On the opposite page of the note-book, facing the notes, 
there should be ruled a vertical line through the center of the 
page, and the customary process of recording the objects 
encountered on the traverse line is to use this vertical line as 
the line of the traverse. Beginning at the bottom of the page, 
plat the first station, A ; then, at the proper distance above 
A , plat, still on the center of the line and disregarding the 
deflections, the second station, B. Crossing this line of the 
traverse, note the topographic features, as streams, roads, 
houses, etc. Where topographic notes are taken in detail 
it is practically impossible to keep a proper record of the 
traverse by considering it as a straight line; in which case, 


210 


TRAVERSE INSTRUMENTS AND METHODS . 


instead of using a central line as the traverse line, an irregular 
line should be drawn up the page, each tangent or deflection 

line of the traverse making an angle with 
the last, which shall agree as nearly as 
possible, by eye estimation or by platting 
with horn protractor, with the angle 
made on the ground. By this means 
the topography of the country will not 
be distorted in recording it on the sketch 
page (Fig. 68). 

88. Platting Transit Notes with 
Protractor and Scale. —Transit notes 
may be platted in two ways: 

1. By means of a protractor and 
scale, and 

2. By latitudes and departures (Art. 

9 °). 

In platting with a protractor (Art. 
Transit Road Tra- 89) and scale, set the center of the pro- 
VERSE - tractor over the occupied station as 

platted on the map, set the zero on the prolongation of the 
last sight, and plat off the deflection to right or left by the 
proper number of degrees. Then, removing the protractor, 
plat on this new deflection line the proper distance to scale. 
This deflection line should be drawn sufficiently long, so that 
when the protractor is centered over the second station this 
old deflection line will appear on the map as the zero-point 
on which to set the protractor for the next following deflection. 

89. Protractors. —In the platting of traverses run with a 
prismatic compass, the simplest form of a semicircular horn 
protractor will fill the requirements; also in platting recon¬ 
naissance triangulation in order to determine the relative 
positions of stations. Where any attempt is made at accu¬ 
rate platting, as of traverse run with transit, a full-circle 
vernier arm protractor should be used (Fig. 69). Where 



Fig. 68. — Plat of 


PROTRA CTORS. 


21 I 


angle-reading instruments are used in topographic surveying, 
it is expected that the work done will be of such high quality 



Fig. 69.—Full-circle Vernier Protractor. 


as to call for computation either of latitudes and departures, 
in the case of traverse, or of geodetic coordinates, in primary 
triangulation (Chaps. XXIV and XXIX). 

Occasionally the topographer, especially if engaged in 
hydrographic surveying, will need to locate his position by 
the three-point problem (Art. 74), that is, by angles read 
from an unknown to three known positions. The location of 



Fig. 70.—Three-arm Protractor. 

his unknown and occupied point may be computed (Art. 
263), or it may be platted graphically by means of an instru- 













































































































212 


TRAVERSE INSTRUMENTS AND METHODS. 


ment known as the three-armed protractor, which is very 
useful and does excellent work of this kind. (Fig. /O.) 

90. Platting Transit Notes by Latitudes and Depart¬ 
ures. —This is the most accurate method of platting transit 
notes, and is identical with that employed in platting traverse 
run with theodolite for primary control (Chap. XXIV). The 
more common expression ‘‘departures” refers to easting 
and westing, known in astronomical phraseology as “ longi¬ 
tude.” The computing and platting are not done with the 
same care and accuracy as for primary traverse. The con¬ 
vergence of the meridians is rarely recorded, nor are the errors 
of measurement of deflection angles corrected by astronomical 
azimuths or by checks on known geodetic positions. 

The process consists in platting by rectangular coordinates 
to reference lines which are drawn at right angles and cor¬ 
respond approximately to latitude and longitude lines. The 
horizontal line is assumed as the initial latitude, and is the 
zero from which differences of latitude are measured up and 
down. In other words, it is the line of abscissae, and along it 
are measured off the differences of longitude or departure 
from the vertical line, which is zero of longitudes. All north¬ 
ings on the traverse line are measured upwards and all south¬ 
ings downwards, and they are denoted by the signs -|- and —. 
Eastings and westings, respectively, are measured to the right 
or left of the vertical line or initial longitude line, and are 
denoted also by signs -f- and —-. 

The zero of azimuth is, as in geodetic computation 
(Art. 285), supposed to be at the south, while the north 
is 180 0 . The azimuth is measured in the same direction 
as the motion of the hands of a watch, 90° being to the 
west and 270° to the east. A simple manner of keeping 
signs in mind during computation is from inspection of a 
diagram similar to Fig. 71. 

The total latitudes and departures are computed only for 
important points, as crossings of roads, streams, etc. The 


PLATTING TP A NS IT NOTES. 


213 


intermediate bends in the road traverse may be platted by 
protractor. The total latitudes and departures are deter¬ 
mined for each governing point by summation of the partial 


N 

190' 



270 E 


Fig. 71.—Signs of Latitudes and Departures. 
latitudes and departures to that point. They are derived by 
two methods. (1) By adding to the logarithms of the distances 
(Table V) the logarithms of the sines of the azimuths (Table 
VI). The total departures are obtained by adding to the 
logarithms of the same distances the logarithms of the cosines of 
the corresponding azimuths. The second method of comput¬ 
ing latitudes and departures is by means of a table of natural 
functions (Tables XI and XII). 

COMPUTATION OF LATITUDES AND DEPARTURES. 



To Station 

59 +• 

To Station 

7 1 +• 

To Station 

78 +. 

Log. sin. (Dep.).. 

9-5590 

9.6394 

9-0950 

‘ dist. 

2.6385 

3.0719 

2.8837 

“ dep. 

2-1975 

2 .7113 

I.9787 

Departure (feet).. 

157-5 

514-3 

9 52 

Log. cosin (Lat.). 

9.9694 

9-9543 

9.9966 

“ dist. 

2.6385 

2.0719 

2.8837 

“ dep. 

2.6079 

2.0262 

2.8803 

Departure (feet).. 

405-4 

106.3 

759-2 





























2 14 


TRAVERSE INSTRUMENTS AND METHODS. 


The details of the computation are given in extenso in 
Chapter XXIV, for primary traverse. The foregoing ex¬ 
ample is given here, however, as an illustration of the simpler 
mode of computing latitudes and departures from transit 
notes, and is taken from the example of such notes given in 
Article 87. 

The table of four-place logarithms of numbers on pages 
215 and 216 is derived from Prof. J. B. Johnson’s “Theory 
and Practice of Surveying”; that of similar trigonometric 
functions on pages 2 17 to 22 1 is from Gauss’ well-known tables. 
By their use a traverse run with engineer’s transit can be 
computed by latitudes and departures with sufficient accuracy. 

91. Prismatic Compass. —This is a useful instrument for 
determining directions on reconnaissance traverses of roads, 
streams, etc. It is unnecessary to mount it on a Jacob’s- 
staff or tripod, as it is easily read while held in the hand. It 
has a full circle of 360 degrees (Fig. 72) and folded sights. 



Fig. 72.—Prismatic Compass. 


The foresight has a cross-hair, and the rear- or eye-sight is so 
provided with a prism that while the instrument is pointed 










































































LOGARITHMS OF NUMBERS, 


215 


Table V. 

LOGARITHMS OF NUMBERS. 


c/5 

u 

a; 











Proportional Parts. 

£ 

E 

£ 

0 

1 

2 

3 

4 

5 

0 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

IO 

.OOOO 

.0043 

.0086 

.0128 

.OI7O 

.0212 

.0253 

.0294 

• 0.334 

.0374 

4 

8 

12 

17 

21 

25 

29 

33 

37 

I I 

.0414 

•0453 

.0492 

• o 53 x 

.056Q 

.0607 

.0645 

.0682 

.0719 

•0755 

4 

8 

IX 

15 

x 9 

2 3 

26 

30 

34 

12 

.0792 

.0828 

.0864 

.0899 

• 0934 

.0969 

. 1004 

.1038 

. 1072 

. 1106 

3 

7 

10 

14 

x 7 

21 

24 

28 

3 1 

13 

• IX 39 

•H 73 

. 1206 

.1239 

. I27I 

• 1303 

•1335 

■ 1367 

• I 399 

.1430 

3 

6 

10 

x 3 

l6 

19 

23 

26 

29 

x 4 

. 1461 

.1492 

•1523 

•1553 

.1584 

.1614 

.1644 

•1673 

• I 7°3 

• x 732 

3 

6 

9 

12 

15 

18 

21 

2 4 

27 

15 

.1761 

.1790 

.18x8 

.1847 

• 1875 

.1903 

•i 93 i 

• x 959 

.1987 

.2014 

3 

6 

8 

II 

14 

x 7 

20 

22 

25 

l6 

.2041 

.2068 

•2095 

.2122 

.2148 

• 2175 

.2201 

.2227 

.2253 

.2279 

3 

5 

8 

II 

x 3 

l6 

18 

21 

24 

x 7 

.2304 

.2330 

•2355 

• 2380 

.2405 

.2430 

• 2455 

.2480 

.2504 

.2529 

2 

5 

7 

IO. 

12 

x 5 

x 7 

20 

22 

18 

•2553 

•2577 

.2601 

• 2625 

. 2648 

. 2672 

.2695 

.2718 

.2742 

.2765 

2 

5 

7 

9 

12 

t 4 

l6 

x 9 

21 

x 9 

.2788 

.2810 

•2833 

.2856 

.2878 

. 29OO 

• 2923 

•2945 

.2967 

.2989 

2 

4 

7 

9 

I 1 

x 3 

16 

18 

20 

20 

• 3 OI ° 

• 3 ° 3 2 

•3054 

• 3075 

.3096 

• 3 118 

• 3139 

• 3 i6 ° 

.3181 

.3201 

2 

4 

6 

8 

I 1 

x 3 

x 5 

x 7 

x 9 

21 

.3222 

•3243 

.3263 

.3284 

•3304 

• 33 2 4 

• 3345 

•3365 

•3385 

• 34°4 

2 

4 

6 

8 

to 

12 

x 4 

l6 

18 

22 

•3424 

•3444 

• 3464 

•3483 

• 3502 

•3522 

• 3541 

.3560 

•3579 

•3598 

2 

4 

6 

8 

IO 

12 

x 4 

x 5 

x 7 

23 

• 3 6i 7 

.3636 

• 3 6 55 

•3674 

.3692 

• 37 11 

•3729 

•3747 

.3766 

•3784 

2 

4 

6 

7 

9 

I I 

13 

x 5 

x 7 

24 

.3802 

.3820 

•3838 

•3856 

•3874 

.3892 

• 39°9 

•3927 

•3945 

.3962 

2 

4 

5 

7 

9 

I I 

12 

x 4 

l6 

25 

•3979 

•3997 

.4014 

.4031 

. 4048 

.4065 

.4082 

.4099 

.4116 

• 4 T 33 

2 

3 

5 

7 

9 

IO 

12 

x 4 

x 5 

26 

.4150 

.4166 

.4183 

. 42OO 

.4216 

.4232 

•4249 

.4265 

.4281 

.4298 

2 

3 

5 

7 

8 

IO 

11 

x 3 

x 5 

27 

• 43 x 4 

• 433 ° 

• 4346 

.4362 

• 4378 

•4393 

.4409 

•4425 

.4440 

.4456 

2 

3 

5 

6 

8 

9 

II 

x 3 

x 4 

28 

.4472 

.4487 

.4502 

.4518 

• 4533 

•4548 

• 45 6 4 

•4579 

•4594 

.4609 

2 

3 

5 

6 

8 

9 

II 

12 

x 4 

29 

.4624 

•4639 

•4654 

. 4669 

.4683 

.4698 

47 i 3 

•4728 

•4742 

•4757 

1 

3 

4 

6 

7 

9 

IO 

12 

x 3 

3 ° 

• 477 x 

.4786 

.4800 

.4814 

.4829 

•4843 

•4857 

00 

.4886 

.4900 

I 

3 

4 

6 

7 

9 

10 

11 

x 3 

3 X 

.4914 

.4928 

• 4942 

• 4955 

.4969 

■4983 

•4997 

• 5 °n 

.5024 

.5038 

I 

3 

4 

6 

7 

8 

IO 

11 

12 

32 

•5051 

.5065 

• 5079 

.5092 

• 5105 

• 5 XX 9 

•5132 

• 5 i 45 

• 5 I 59 

•5172 

I 

3 

4 

5 

7 

8 

9 

11 

12 

33 

•5185 

.5198 

.5211 

.5224 

•5237 

•5250 

.5263 

.5276 

•5289 

• 53 ° 2 

1 

3 

4 

5 

6 

8 

9 

IO 

12 

34 

• 53 x 5 

.5328 

•5340 

•5353 

.5366 

•5378 

• 539 i 

•5403 

.5416 

.5428 

I 

3 

4 

5 

6 

8 

9 

10 

II 

35 

•5441 

•5453 

• 5463 

• 5478 

• 5490 

• 5502 

•5514 

•5527 

•5539 

• 555 1 

I 

2 

4 

5 

6 

7 

9 

IO 

II 

36 

•5563 

•5575 

•5587 

• 5599 

.56x1 

.5623 

•5635 

•5647 

.5658 

.5670 

I 

2 

4 

5 

6 

7 

8 

IO 

11 

37 

.5682 

•5694 

•5705 

• 5717 

• 5729 

•5740 

•5752 

•5763 

•5775 

•5786 

I 

2 

3 

5 

6 

7 

8 

9 

IO 

38 

•5798 

.5809 

.5821 

• 5832 

•5843 

•5855 

.5866 

•5877 

.5888 

•5899 

I 

2 

3 

5 

6 

7 

8 

9 

IO 

39 

• 59 xx 

.5922 

•5933 

• 5944 

•5955 

.5966 

•5977 

.5988 

•5999 

.6010 

I 

2 

3 

4 

5 

7 

8 

9 

IO 

40 

.6021 

.6031 

.6042 

• 6053 

. 6064 

.6075 

.6085 

. 6096 

. 6107 

.6117 

I 

2 

3 

4 

5 

6 

8 

9 

IO 

4 i 

.6128 

.6138 

.6149 

.6160 

.6170 

.6180 

.6191 

.6201 

.6212 

.6222 

I 

2 

3 

4 

5 

6 

7 

8 

9 

42 

.6242 

.6243 

.6253 

.6263 

.6274 

.6284 

.6294 

.6304 

.6314 

•6325 

I 

2 

3 

4 

5 

6 

7 

8 

9 

4 J 

•6335 

•6345 

•6355 

• 6 3 6 5 

• 6375 

.6385 

•6395 

.6405 

.6415 

.6425 

I 

2 

3 

4 

5 

6 

7 

8 

9 

44 

•6435 

.6444 

•6454 

. 6464 

.6474 

.6484 

•6493 

• 6503 

•6513 

.6522 

I 

2 

3 

4 

5 

6 

7 

8 

9 

45 

.6532 

.6542 

•6551 

.6561 

• 657' 

.6580 

.6590 

•6599 

.6609 

.6618 

I 

2 

3 

4 

5 

6 

7 

8 

9 

46 

.6628 

.6637 

.6 646 

.6656 

.6665 

.6675 

.6684 

.6693 

.6702 

.6712 

I 

2 

3 

4 

5 

6 

7 

7 

8 

47 

.6721 

.6730 

•6739 

.67491 

.6758 

.6767 

.6776 

.6785! 

.6794 

.6803 

I 

2 

3 

4 

5 

5 

6 

7 

8 

48 

.6812 

.6821 

.6830 

.6839 

.6848 

.6857 

.6866 

.6875 

.6884 

.6893 

I 

2 

3 

4 

4 

5 

6 

7 

8 

49 

.6902 

.69x1 

. 6920 

.6928 

• 6937 

. 6946 

•6955 

.6964 

.6972 

.6981 

I 

2 

3 

4 

4 

5 

6 

7 

8 

50 

.6990 

.6998 

. 7007 

. 7016 

.7-024 

• 7°33 

.7042 

.7050 

•7059 

. 7067 

1 

2 

3 

3 

4 

5 

6 

7 

8 

5 i 

. 7076 

•7084 

.7093 

.7IOI 

.7110 

.7118 

. 7126 

• 7 i 35 

• 7 i 43 

• 7 X 52 

1 

2 

3 

3 

4 

5 

6 

7 

8 

52 

.7x60 

.7168 

• 7 X 77 

• 7185 

•7193 

. 7202 

. 7210 

. 7218 

. 7226 

•7235 

I 

2 

2 

3 

4 

5 

6 

7 

7 

53 

•7243 

• 7251 

•7259 

. 7267 

•7275 

.7284 

. 7292 

.7300 

.7308 

• 73 i 6 

I 

2 

2 

3 

4 

5 

6 

6 

7 

54 

•7324 

•7332 

•7340 

• 7348 

• 735 ^ 

• 73 6 4 

•7372 

.7380 

.7388 

.7396 

I 

2 

2 

3 

4 

5 

6 

6 

7 

































































2l6 


TRAVERSE INSTRUMENTS AND METHODS 


Table V. 

LOGARITHMS OF NUMBERS. 


X 

i- 












— 

Proportional Parts. 

s 

£ 

5 

£ 

0 

I 

2 

3 

4 

5 

G 

7 

8 

9 

1 

3 

3 

4 

5 

G 

7 

1 

8 

[ "" 

9 

55 

.7404 

.7412 

.7419 

.7427 

•7435 

•7443 

• 745 i 

•7459 

.7466 

•7474 

I 

2 

2 

3 

4 

5 

5 

6 

7 

5 6 

.7482 

.7490 

•7497 

•7505 

•7513 

.7520 

.7528 

•7536 

•7543 

•/ 55 i 

I 

2 

2 

3 

4 

5 

5 

6 

7 

57 

•7559 

.7566 

•7574 

.7582 

•7589 

•7597 

.7604 

. 7612 

.7619 

. 7627 

I 

2 

2 

3 

4 

5 

5 

6 

7 

58 

•7634 

.7642 

.7649 

•7657 

.7664 

.7672 

.7679 

.7686 

.7694 

.7701 

I 

I 

2 

3 

4 

4 

5 

6 

7 

59 

.7709 

.7716 

•7723 

• 773 1 

•7738 

•7745 

•7752 

.7760 

.7767 

•7774 

I 

I 

2 

3 

4 

4 

5 

6 

7 

60 

. 7782 

•7789 

.7796 

.7803 

. 7810 

.7818 

•7825 

•7832 

•7839 

.7846 

I 

I 

2 

3 

4 

4 

5 

6 

6 

61 

■7853 

. 786c 

.7868 

•7875 

.7882 

.7889 

. 7896 

•7903 

.791° 

.7917 

I 

I 

2 

3 

4 

4 

5 

6 

6 

62 

.7924 

• 793 i 

.7938 

• 7945 

•7952 

•7950 

. 7966 

•7973 

.7980 

•7987 

I 

I 

2 

3 

3 

4 

5 

6 

6 

63 

•7993 

. 8000 

. 8007 

. 8014 

. 8021 

.8028 

• 80 ^5 

.8041 

.8048 

•8055 

I 

I 

2 

3 

0 

J 

4 

5 

5 

6 

64 

.8062 

. 8069 

•8075 

. 8082 

.8089 

.8096 

.8102 

. 8109 

.8116 

.8122 

I 

I 

2 

3 

3 

4 

5 

5 

6 

65 

.8129 

• 8136 

.8142 

.8149 

.8156 

.8162 

.8169 

.8176 

.8182 

.8189 

I 

I 

2 

3 

3 

4 

5 

5 

6 

66 

.8195 

.8202 

.8209 

.8215 

.8222 

.8228 

•8235 

. 8241 

.8248 

.8254 

I 

I 

2 

3 

3 

4 

5 

5 

6 

67 

. 8261 

.8267 

.8274 

. 8280 

.8287 

.8293 

. 8299 

. 8306 

.8312 

.8319I 

I 

I 

2 

3 

3 

4 

5 

5 

6 

68 

•8325 

•8331 

•8338 

•8344 

•8351 

•8357 

• 8363 

.8370 

.8376 

.8382 

I 

I 

2 

3 

3 

4 

4 

5 

6 

69 

.8388 

•8395 

.8401 

.8407 

.8414 

.8420 

.8426 

.8432 

•8439 

•8445 

I 

I 

2 

2 

3 

4 

4 

5 

6 

70 

.8451 

•8457 

.8463 

.8470 

.8476 

.8482 

.8488 

.8494 

.8500 

.8506 

I 

I 

2 

2 

3 

4 

4 

5 

6 

71 

•8513 

.8519 

.8525 

•8531 

•8537 

•8543 

•8549 

•8555 

.8561 

.8567 

I 

I 

2 

2 

3 

4 

4 

5 

5 

72 

•8573 

‘ -8579 

■8585 

.8591 

•8597 

.8603 

.8609 

.8615 

. 8621 

. 8627 

I 

I 

2 

2 

3 

4 

4 

5 

5 

73 

•8633 

.8639 

.8645 

• 8651 

■8657 

.8663 

.8669 

.8675 

.8681 

.8686 

I 

1 

2 

2 

0 

0 

4 

4 

5 

5 

74 

.8692 

.8698 

.8704 

.8710 

.8716 

.8722 

•8727 

•8733 

•8739 

•8745 

I 

I 

2 

2 

3 

4 

4 

5 

5 

75 

• 875 > 

.8756 

.8762 

.8768 

•8774 

.8779 

•8785 

.8791 

•8797 

.8802 

I 

I 

2 

2 

3 

3 

4 

5 

5 

76 

.8808 

.8814 

.8820 

.8825 

•8831 

.8837 

. 8842 

.8848 

• 8854 

• 8859 

I 

I 

2 

2 

3 

3 

4 

5 

5 

77 

.8865 

.8871 

.8876 

.8882 

.8887 

.8893 

. 8899 

.8904 

.8910 

.8915 

I 

I 

2 

2 

3 

3 

4 

4 

5 

78 

.8921 

.8927 

.8932 

•8938 

•8943 

• 8949 

•8954 

.8960 

.8965 

.8971 

I 

I 

2 

2 

3 

3 

4 

4 

5 

79 

.8976 

.8982 

.8987 

•8993 

.8998 

.9004 

.9OO9 

.9015 

.9020 

.9025 

I 

I 

2 

2 

3 

3 

4 

4 

5 

80 

.9031 

.9036 

.9042 

.9047 

•9053 

.9058 

.9063 

.9069 

.9074 

.9079 

I 

I 

2 

2 

3 

3 

4 

4 

5 

81 

.9085 

.9O9O 

. 9096 

.9101 

.9106 

.9112 

.9117 

.9122 

.9128 

• 9 1 33 

I 

I 

2 

2 

3 

3 

4 

•4 

5 

82 

• 9 i 38 

• 9 I 43 

.9149 

• 9 I 54 

• 9 r 59 

.9165 

.9170 

• 9 I 75 

.9180 

.9186 

I 

I 

2 

2 

3 

3 

4 

4 

5 

83 

.9191 

.9196 

.9201 

.9206 

.9212 

.9217 

. 9222 

.9227 

• 9232 

.9238 

I 

I 

2 

2 

3 

3 

4 

4 

5 

84 

• 9 2 43 

.9248 

•9253 

•9258 

.9263 

.9269 

.9274 

•9279 

.9284 

.9289 

I 

I 

2 

2 

3 

3 

4 

4 

5 

85 

.9294 

.9299 

•9304 

•9309 

•9315 

.9320 

•9325 

•9330 

•9335 

•9340 

I 

I 

2 

2 

3 

3 

4 

4 

c 

86 

•9345 

•9350 

•9355 

.9360 

•9365 

•9370 

•9375 

.9380 

•9385 

•9390 

I 

I 

2 

2 

3 

3 

4 

4 

5 

87 

•9395 

. 9400 

•9405 

• 94 IQ 

•9415 

.9420 

•9425 

•9430 

•9435 

.9440 

O 

I 

X 

2 

2 

3 

3 

4 

4 

88 

•9445 

• 9450 

•9455 

.9460 

•9465 

.9469 

•9474 

•9479 

•9484 

•9489 

0 

I 

I 

2 

2 

3 

3 

4 

4 

89 

•9494 

•9499 

.9504 

•9509 

• 95 i 3 

.9518 

•9523 

•9528 

•9533 

•9538 

O 

1 

I 

2 

2 

3 

3 

4 

4 

90 

.9542 

•9547 

•9552 

•9557 

.9562 

.9566 

•9571 

•9576 

.9581 

.9586 

O 

I 

I 

2 

2 

3 

3 

4 

4 

9 i 

•9590 

•9595 

.9600 

.9605 

.9609 

.9614 

.9619 

.9624 

.9628 

• 96.33 

O 

I 

I 

2 

2 

3 

3 

4 

4 

92 

.9638 

•9643 

.9647 

.9652 

• 9 6 57 

.9661 

.9666 

.9671 

•9675 

.9680 

O 

I 

I 

2 

2 

3 

3 

4 

4 

93 

.9685 

.9689 

.9694 

.9699 

•9703 

.9708 

•9713 

.9717 

.9722 

•9727 

O 

I 

1 

2 

2 

3 

3 

4 

4 

94 

• 973 i 

•9736 

•9741 

•9745 

•9750 

•9754 

•9759 

■9763 

.9768 

•9773 

O 

I 

I 

2 

2 

3 

3 

4 

4 

95 

•9777 

.9782 

.9786 

.9791 

•9795 

.9800 

• 9805 

.9809 

.9814 

.9818 

O 

I 

I 

2 

2 

3 

3 

4 

4 

96 

.9823 

.9827 

.9832 

.9836 

.9841 

• 9845 

.9850 

•9854 

•9859 

.9863 

O 

I 

I 

2 

2 

3 

3 

4 

4 

97 

.9868 

.9872 

.9877 

.9881 

.9886 

.9890 

•9894 

•9899 

•9903 

.9908 

O 

I 

I 

2 

2 

3 

3 

4 

4 

98 

.9912 

.9917 

.9921 

.9926 

•9930 

•9934 

•9939 

'99431 

.9948 

•9952 

O 

I 

I 

2 

2 

3 

3 

4 

4 

99 

• 995 ^ 

.9961 

• 99 6 5 

.9969 

•9974 

.9978 

, -9983 

•9987 

.9991 

.9996 

° 

I 

I 

2 

2 

3 

3 

3 

4 

I 

































































LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 2\J 


Table VI. 

LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 


0 / 

L. Sin. 

0 0 

— 

10 

7-4637 

20 

7.7648 

3 ° 

7.9408 

40 

8.0658 

5 ° 

8.1627 

1 O 

8.2419 

IO 

8.3088 

20 

8.3668 

30 

8.4179 

40 

8-4637 

5 ° 

8.4050 

2 0 

8.5428 

10 

8.5776 

20 

8.6097 

3 ° 

8.6397 

40 

8.6677 

50 

8.6940 

3 0 

8.7188 

IO 

8.7423 

20 

8.7645 

30 

8-7857 

40 

8.8059 

50 

8.8251 

4 0 

8.8456 

10 

8.8613 

20 

8.8783 

3 ® 

8.8946 

40 

8.9104 

50 

8.9256 

5 0 

8.9403 

10 

8.9545 

20 

8.9682 

3 ° 

8.9816 

40 

8.9945 

50 

9.OO7O 

6 0 

9.0192 

10 

9.0311 

20 

9.0426 

20 

9-0530 

40 

9.0648 

50 

9-0755 

7 0 

9.0859 

10 

9.0961 

20 

9.1060 

3 ° 

9* 1x 57 

40 

9• 1 252 

5 ° 

9-1345 

8 0 

9 M 36 

IO 

9-1525 

20 

9.1612 

3 ° 

9.1697 

40 

9.1781 

5 ° 

9.1863 

9 0 

9- I 943 

IO 

9.2022 

20 

9.2IOO 

3 ° 

9.2176 

40 

9.2251 

50 

9.2324 

IO 0 

9-2397 


L. Cos. 


d. 


3011 

1760 

1250 

969 

792 

669 
580 
5 11 
458 
4 i 3 
378 

348 

321 

300 

280 

263 

248 

2 35 

222 

212 

202 

192 

185 

177 

170 

163 

158 

J 52 

M 7 

142 

137 

134 

129 

125 

122 

XI 9 

XI 5 

1X 3 

109 

107 

104 

102 

99 

97 

95 

93 

9 1 

89 

87 

85 

84 

82 

80 

79 

78 

76 

75 

73 

73 


L.Tang 


7-4637 

7 •7648 

7.9409 

8.0658 

8.1627 

8.2419 

8.3089 

8.3669 
8.4181 
8.4638 

8.5053 

8 . 543 x 

8-5779 
8.6101 
8.6401 
8.6682 
8.6945 

8.7194 

8.7429 

8.7652 
8.7865 

8.8067 

8.8261 

8.8446 

8.8624 

8.8795 

8.8960 
8.9118 
8.9272 

8.9420 

8.9563 

8.9701 

8.9836 

8.9966 

9.0093 

9.0216 

9.0336 

9.0453 

9.0567 

9.0678 

9.0786 

9.0891 

9.0995 
9.1096 
9.1194 
9.1291 

91385 

9.1478 

9-1569 

9.1658 

9- x 745 

9.1831 

9-1915 

9.1997 

9.2078 
9.2158 
9.2236 

9-2313 

9.2389 


d. 


9.2463 


L Cotg. 


d. c. 

L.Cotg. 

L. Cos. 



— 

O.OOOO 

0 90 

3011 

2-5363 

O.OOOO 

50 

1761 

2.2352 

0.OOOO 

40 

1249 

2.0591 

0.OOOO 

30 

969 

792 

670 

580 

1-9342 

0.0000 

20 

1.8373 

0.ocoo 

IO 

1.7581 

9.9999 

089 

1.6911 

9.9999 

50 

512 

1.6331 

9.9999 

40 

4^7 

1.5819 

9•9999 

30 

415 

1.5362 

9.9998 

20 

378 

348 

322 

300 

281 

1-4947 

9.9998 

IO 

1.4569 

9.9997 

088 

1.4221 

9.9997 

50 

1-3899 

9.9996 

40 

1-3599 

9.9996 

3 ° 

263 

1-3318 

9-9995 

20 

1•3055 

9-9995 

IO 

249 

1.2806 

9-9994 

0 87 

2 35 

223 

213 

1-2571 

9-9993 

5 ° 

1.2348 

9.9993 

40 

202 

1-2135 

9.9992 

30 

194 

185 

178 

171 

165 

1-1933 

9-9991 

20 

1.1739 

9.9990 

IO 

1.1554 

o.9989 

08G 

1.1376 

9.9989 

50 

1.1205 

9.9988 

40 

158 

154 

I.TO4O 

9.9987 

30 

1.0882 

9.9986 

20 

1.0728 

9.9985 

IO 

148 

1.0580 

9.9983 

0 8£> 

143 

138 

1-0437 

9.9982 

50 

135 

130 

I.0299 

9.0981 

40 

1.0164 

9.9980 

3 ° 

127 

1.0034 

9.9979 

20 

0.9907 

9.9977 

IO 

123 

120 

H7 

0.9784 

9.9976 

0 84 

0.9664 

9-9975 

50 

H4 

0-9547 

9-9973 

40 

in 

0-9433 

9.9972 

30 

108 

0.9322 

9.9971 

20 


0.9214 

9-99 9 

IO 

105 

O.giOQ 

9.9968 

0 83 

104 

101 

0.9005 

9.9966 

50 

98 

97 

94 

0.8904 

9.9964 

40 

0.8806 

9.9963 

30 

0.8709 

9.9961 

20 

0.8615 

9-9959 

IO 

93 

0.8522 

9.9958 

o 82 

9 1 

89 

0.8431 

9.9956 

50 

87 

0.8342 

9-9954 

40 

86 

08255 

9-9952 

30 

84 

Q 0 

0.8169 

9.9950 

20 

0.8085 

9.9948 

IO 

02 

0 T 

0.8003 

0.9946 

08I 

ol 

80 

0.7922 

9.9944 

50 

78 

0.7842 

9.9942 

40 

77 

79 

0.7764 

9.9940 

30 

0.7687 

9.9938 

20 

0.7611 

9.9936 

10 

70 

0-7537 

9.9934 

| 08O 

] 

. d. c. 

L.Tang 

L. Sin. 

1 ' 0 


10 

20 

3® 

40 

5° 

<»0 

70 

80 

90 

IOO 
I IO 

120 

* 3 ° 

140 

T 5 ° 

160 

170 

180 

190 

200 

210 

220 

270 

240 

250 

260 

270 

280 

290 


s. 

T. 

— 

— 

6.4637 

6.4637 

6.4637 

6.4637 

6.4637 

6.4637 

6-4637 

6.4637 

6.4637 

6.4678 

6.4637 

6.4638 

6.4637 

6.4637 

6.4637 

6.4637 

64637 

6.4638 

6.4638 

6.4638 

6.4638 

6.4639 

6.4636 

6.4639 

6.4636 

6.4636 

6.4636 

6.4636 
6.4635 

6.4639 

6.4640 
6.4640 

6.4640 

6.4641 

6.4675 

6.4641 

6.4635 

6.4635 

6.4635 

6.4634 

6•4634 

6.4642 

6.4642 

6.4643 

6.4643 

6.4644 

6.4634 

6.4644 

6.4633 

6.4633 

6.4633 

64632 

6.4632 

6.4645 

6.4646 
6.4646 

64647 

6 4648 


o° II . 5 ° 

++I 1 

be^cc *¥ 

II !l be fcjr 

« be || II 

s 3 %, 

i/j —' 

be be bo 

000 


85 ° II- 9 o° 
c/3 E- 1 c/2 E -1 
H—h I I 


d es 
I 


fcuc 

05 *-* 

o o 

o 0 o O 

O' o> be be 
— o o 
be b tr“ 

S.S. || || 
II11^ 

2s-b 

o o g 
o u c? 

be be be 

000 




p 

] 

D 





14*2 

13 4 

I 

134 

l 

*29 

1 

14.2 

13 . 

7 

'3 

■4 

I 

2.9 

2 

28.4 

27 . 

4 

26.8 

2 

5-8 

3 

42.6 

41 • 

I 

40.2 

38.7 

4 

56.8 

54- 

8 

53-6 


, 1.6 

5 

71 .O 

68 . 

5 

67.0 

64-5 

6 

85.2 

82 . 

2 

8 o. 4 

77-4 

7 

99.4 

95- 

9 

93-8 

90.3 

8 

113.6 

IO 9 . 

6 

107.2 

103.2 

9 

127.8 

123 

3 

120.6 

116.1 


125 

122 

119 

1 

15 

I 

12.5 

12 . 

2 

I 

•9 

11 -5 

2 

25.0 

24 . 

4 

23.8 

2 3 . 0 

3 

37-5 

56 . 

6 

35-7 

34-5 

4 

50.0 

48 

8 

47.6 

46.0 

5 

62.5 

61 

O 

59-5 

57-5 

6 

75-o 

73 

2 

7 

[.4 

69.0 

7 

87.5 

85 

4 

8 - 

? *3 

80.5 

8 

IOO. O 

97 

6 

95-2 

92.0 

9 

112.5 

109 

8 

IO 7 . T 

103-5 


113 

109 

1 

0 7 

104 

10*2 

I 

11 .3 

10*9 

10.7 

IO 

4 

10.2 

2 

22 .6 

21.8 

21.4 

20 

.8 

20.4 

3 

33-9 

32.7 

32.1 

31 

.2 

30.6 

4 

45.2 

43-6 

42.8 

41 

.6 

40.8 

5 

56-5 

54-5 

53-5 

52 

.0 

51.0 

6 

67.8 

65-4 

64.2 

62 

•4 

61.2 

7 

79.1 

70-3 

74-9 

72 

.8 

71.4 

8 

90.4 

87.2 

85.6 

83 

.2 

81.6 

9 

IOI .7 

98.1 

96.3 

93 

.6 

91.8 


P. P. 








































































































































































218 


TRA VERSE INSTRUMENTS AND METHODS , 


Table VI. 

LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 


0 / 

L. Sin. 

d. 

L.Tang 

10 0 

9.2397 


9.2463 

10 

9.2468 

7 1 

70 

68 

9-2536 

20 

9.2538 

9.2609 

3° 

9.2606 

68 

9.2680 

401 

9.2674 

66 

9.2750 

501 

9.2740 

66 

64 

9.2819 

1 1 0! 

9.2806 

9.2887 


9.2870 


IO 

64 

9-2953 

20 

9.2934 

63 

Q.3020 

30 

9.2997 

6l 

9•3085 

40 

9.3058 

6l 

9-3149 

50 

9 - 3 IX 9 

60 

9.3212 

12 0 



9-3179 

59 

58 

9327s 

IO 

9-3238 

9-3336 

20 

9 • 329 6 

57 

9-3397 

3 ° 

9-3353 

57 

9-3458 

40 

9.3410 

56 

55 

54 

54 

9 3517 

5 ° 

9.3466 

9-3576 

13 0 

9-3521 

9-3634 

IO 

9-3575 

9.3691 

20 

9.3629 

53 

9-3748 

30 

9.3682 

52 

9.3804 

40 

9-3734 

52 

9-3859 

5 ° 

9.3786 

9 - 39*4 

14 0 

9-3837 

5 1 

50 

50 

9.3968 

IO 

9.3887 

9.4021 

20 

9-3937 

49 

9.4074 

30 

40 

9-3986 

9 • 4°35 

49 

48 

9 4127 

9.4178 

50 

9.4083 

9.4230 

15 0 

9.4/30 

47 

9.4281 



47 


IO 

9-4177 

46 

9-4331 

20 

9.4223 

46 

9-4381 

3 ° 

9.4269 

45 

9.4430 

40 

9 - 43 I 4 

45 

44 

44 

44 

9.4479 

50 

9-4359 

9-4527 

16 0 

9.4403 

9-4575 

IO 

9.4447 

9.4622 

20 

9.4491 

42 

9.4669 

3 ° 

9-4533 

43 

9 - 47 i 6 

40 

9-4576 

42 

4 i 

4 i 

4 i 

9.4762 

50 

9 4618 

9.4808 

17 0 

9-4659 

9-4853 

IO 

9.4700 

9.4898 

20 

9.4741 

4 ° 

9-4943 

30 

9.4781 

40 

9.4987 

40 

9.4821 

4 ° 

9-5031 

50 

9.4861 

39 

39 

38 

9-5075 

18 0 

9.4900 

9 - 5 ii 8 

IO 

9-4939 

9 - 5 i 6 i 

20 

9-4977 

38 

9.5203 

3 ° 

9 - 5 oi 5 

37 

9-5245 

40 

9-5052 

38 

9-5287 

50 

9.5090 

36 

37 

36 

9-5329 

19 0 

9.5126 

9-5370 

IO 

9-5163 

9 - 54 H 

20 

9-5199 

36 

9-5451 

3 ° 

9.5235 

35 

9-5491 

40 

9.5270 

36 

9-5531 

5 ° 

9.5306 

35 

9-5571 | 

0 

C 

N 

9-5341 

9 - 56 ii , 


L. Cos. 

d. 

L.Cotg. 


d. c. 


73 

7 i 

70 

69 

68 

66 

67 

65 

64 

6 3 

63 

61 

61 

61 

59 

59 

58 

57 

57 

56 

55 

55 

54 

53 

53 

53 

5 1 

52 

5 i 

5 ° 

50 

49 

49 

48 

48 

47 

47 

47 

46 

46 

45 

45 

45 

44 

44 

44 

43 

43 

42 

42 

42 

42 

41 

41 

40 

40 

40 

40 

40 


d. c. 


L.Cotg. 

L. Cos. 

d. 


0-7537 

9.9934 


0 80 

0.7464 

9-9931 


50 

o. 739 i 

9.9929 


40 

0.7320 

9.9927 


30 

0.7250 

9.9924 


20 

0.7181 

Q.9922 


IO 

0.7113 

9.9919 


0 79 

0.7047 

9.9917 


50 

0.6980 

9.9914 


40 

0.6915 

Q.QQI 2 


30 

0.6851 

9.9909 


20 

0.6788 

9.9907 


IO 

0.6725 

9.9904 


0 

00 

0.6664 

9.9901 


50 

0.6603 

9.9899 


40 

0.6542 

9.9896 


30 

0.6483 

9.9893 


20 

0.6424 

9.9890 


IO 

0.6366 

9.9887 


0 77 

0.6309 

9.9884 


50 

0.6252 

9.9881 


40 

0.6196 

9.9878 


3 ° 

0.6141 

9-9875 


20 

0.6086 

9.9872 


IO 

0.6032 

9.9869 


0 76 

o -5979 

9.9866 


50 

0.5926 

9.9863 


40 

0.5873 

9.9859 


30 

0.5822 

9.9856 


20 

0.5770 

9-9853 


IO 

0.5719 

9.9849 


0 75 

0.5669 

9.9846 

3 

50 

0.5619 

9.9843 


40 

0-5570 

9:9839 

4 

30 

0.5521 

9.9836 

3 

20 

0-5473 

9.9832 


IO 

0.5425 

9.9828 

4 

0 74 

0.5378 

9.9825 

3 

50 

0-5331 

9.9821 


40 

0.5284 

9.9817 

4 

30 

0.5238 

9.9814 

3 

20 

0.5192 

9.9810 

4 

IO 

0.5147 

9.9806 

4 

0 73 

0.5102 

9.9802 

4 

50 

0.5057 

9.9798 

4 

40 

0.5013 

9.9794 


3 ° 

0.4969 

9.9790 


20 

0.4925 

9 9786 


IO 

0.4882 

9.9782 

4 

0 73 

0.4839 

9.9778 

4 

4 

50 

0.4797 

9-9774 

4 

40 

0-4755 

9.9770 

z 

30 

o. 47 i 3 

9-9765 

j 

20 

0 4671 

9-976i 


IO 

0.4630 

9 9757 

4 

0 71 

0.4589 

9-9752 

5 

4 

50 

0-4549 

9.9748 

5 

40 

0.4509 

9 9743 

4 

30 

0.4469 

9-9739 

5 

20 

O.4429 

9-9734 


IO 

0.4380 

9.9730 

4 

0 70 

L.Tang 

L. Sin. 

| / 0 


P. P. 


99 

95 

91 

87 

9.9 

9-5 

9 -i 

8.7 

19.8 

19.0 

18.2 

17.4 

29.7 

28.5 

27.3 

26.1 

39-6 

38.0 

36-4 

34-8 

49-5 

47-5 

45-5 

43-5 

59-4 

57 -o 

54-6 

52.2 

69-3 

66.5 

63-7 

60.9 

79.2 

76.0 

72.8 

69.6 

89.1 

85-5 

81.9 

78.3 


84 

8.4 

16.8 

25.2 
33-6 
42.0 
50.4 

58.8 

67.2 

75 6 


80 

78 

7 5 

71 

68 

8.0 

7.8 

7-5 

7 -i 

6 

.8 

16.0 

15.6 

15.0 

14.2 

13 

.6 

24.0 

23-4 

22.5 

21.3 

20 

•4 

32.0 

31 - 2 

30 0 

28.4 

27 

.2 

40.0 

39 -o 

37-5 

35-5 

34 

.0 

48.0 

46.8 

45 -o 

42.6 

40 

.8 

56.0 

54-6 

52-5 

49-7 

47 

.6 

64.0 

62.4 

60.0 

56.8 

54 

•4 

72.0 

70.2 

67-5 

63-9 

61 

.2 



63 

59 

56 

53 

1 

6-3 

5-9 

5-6 

5-3 

2 

12.6 

11. 8 

11.2 

10.6 

3 

18.9 

17.7 

16.8 

15-9 

4 

25.2 

23.6 

22.4 

21.2 

5 

3i-5 

29-5 

28.0 

26.5 

6 

37-8 

35-4 

33-6 

31.8 

7 

44.1 

4 i -3 

39-2 

37 -i 

8 

50.4 

47.2 

44.8 

42.4 

9 

56.7 

53-i 

50.4 

47-7 


49 

4.9 

9.8 

* 4-7 

19.6 

24-5 

29.4 

34-3 

39.2 

44.1 



50 

19 

48 

47 

1 

5 -o 

4.9 

4.8 

4-7 

2 

10.0 

9.8 

9.6 

9.4 

3 

15.0 

14.7 

14.4 

14.1 

4 

20.0 

19.6 

19.2 

18.8 

5 

25.0 

24-5 

24.0 

23-5 

6 

30.0 

29.4 

28.8 

28.2 

7 

35 -o 

34-3 

33-6 

32.9 

8 

40.0 

39-2 

38.4 

37-6 

9 

45.0 

44.1 

43-2 

42.3 


46 

45 

44 

4.6 

4-5 

4.4 

9.2 

9.0 

8.8 

13 8 

t 3-5 

13.2 

18.4 

18.0 

17.6 

23.0 

22.5 

22.0 

27.6 

27.0 

26.4 

32.2 

3 i -5 

30.8 

36.8 

36.0 

35-2 

41.4 

40-5 

39-6 



43 

4-2 

41 

1 

4-3 

4 2 

4.1 

2 

8.6 

8.4 

8.2 

3 

12.9 

12.6 

12.3 

4 

17.2 

16.8 

16.4 

5 

21-5 

21.0 

20.5 

6 

25.8 

25.2 

24.6 

7 

30.1 

29.4 

28.7 

8 

34-4 

33-6 

32.8 

9 

38-7 

37-8 

36.9 


3 

°-3 

0.6 

0.9 

1.2 

i-5 

1.8 

2.1 
2.4 

2.7 

4 

0.4 

0.8 

1.2 

1.6 

2.0 

2.4 

2.8 

3-2 

3.6 


P. P. 


































































































































































LOGARITHMS OF TRIGONOMETRIC 


Table VI. 

LOGARITHMS OF TRIGONOMETRIC 


0 / 

L. Sin. 

<c 

c 

0 

9-5341 

IO 

9-5375 

20 

9.5409 

30 

9-5443 

40 

9-5477 

50 

9 - 55 io 

21 0 

9-5543 

IO 

9-5576 

20 

9.5609 

30 

9.5641 

40 

9-5673 

50 

9.5704 

22 © 

9-5736 

IO 

9-5767 

20 

9-5798 

30 

9.5828 

40 

9-5859 

50 

9.5889 

23 0 

9-5919 

IO 

9-5948 

20 

9-5978 

30 

9.6007 

40 

9.6036 

5 ° 

9.6065 

24 0 

9.6093 

IO 

9.6121 

20 

9.6149 

30 

9.6177 

40 

9■6205 

50 

9.6232 

25 0 

9.6259 

IO 

9.6286 

20 

9-6313 

30 

9.6340 

40 

9.6366 

50 

9•6392 

26 0 

9.6418 

IO 

9.6444 

20 

9.6470 

30 

9-6495 

40 

9.6521 

5 ° 

9.6546 

27 0 

9.6570 

IO 

9-6595 

20 

9.6620 

30 

9.6644 

40 

9.6668 

5 ° 

9.6692 

28 0 

9.6716 

IO 

9.6740 

20 

9.6763 

30 

9.6787 

40 

9.6810 

50 

9-6833 

29 0 

9.6856 

IO 

9.6878 

20 

9.6901 

30 

9.6923 

40 

9.6946 

50 

g.6968 

u 

© 

0 

9.6990 

! L. Cos 


d. 


34 

34 

34 

34 

33 

33 

33 
33 
32 
3 2 

3 1 

3 2 

3 1 

3 1 

3 0 

3 1 

30 

30 

29 

30 

29 

29 

29 

28 

28 

28 

28 

28 

2 7 

2 7 

27 

27 

27 

26 

26 

26 

26 

26 

25 

26 
2 5 

2 4 

25 
25 
24 
24 

24 

24 

24 

23 

24 
23 
23 
23 

22 

23 

22 

23 

22 

22 


d. 


L.Tang 

d. c. 

L.Coig. 

L. Cos 

9.5611 

39 

39 

38 

39 

38 

38 

37 

38 

37 

37 

37 

36 

36 

36 

36 

36 

35 

36 

35 

34 

35 

34 

35 

34 

34 

33 

34 

33 

34 

33 

33 

3 2 

33 

3 2 

33 

3 2 

3 2 

32 

3 1 

3 2 

31 

32 

3 i 

3 i 

31 

31 

30 

31 

30 

3 ° 

' 31 

3 ° 

30 

30 

29 

30 

29 

3 ° 

29 

29 

0.4589 

9-9730 

9-5650 

9.5689 

9 57 2 7 
9.5766 
9.5804 

0.4350 

0.4311 

o. 4 2 73 

o. 4 2 34 

0.4196 

9.9725 

9.9721 

9.9716 

9.9711 

9.9706 

9.5842 

0.4158 

9 9702 

9-5879 

9-5917 

9-5954 

9-5991 

9.6028 

0 4121 
0.4083 

0.4046 

0.4009 
0 - 397 2 

9.9697 
y .9692 
9.9687 
9.9682 

9 9677 

9.6064 

0.3936 

9.9672 

9.6100 

9.6136 

9.6172 

9.6208 

9.6243 

0.3900 
0.3864 
0.3828 
0.3792 
o -3757 

9 9667 
9.9661 
9.9656 

9-9651 

9.9646 

9.6279 

0.3721 

9.9640 

9.6314 

'9-6348 

9-6383 

9.6417 

9.6452 

0.3686 

0.3652 

O.3617 

0.3583 

0.3548 

9.9635 

9.9629 

9.9624 

9.9618 

9.9613 

9.6486 

o- 35 I 4 

q.9607 

Q.6s20 

9-6553 

9.6587 

9.6620 

9.6654 

0.3480 

0-3447 

o- 34 i 3 

0.3380 

0.3346 

9.9602 

9.9596 

9-9590 

9.9584 

9-9579 

9.6687 

0.3313 

9-9573 

9.6720 

9.6752 

9.6785 

9.6817 

9.6850 

0.3280 

0.3248 

0.3215 

0.3183 

0.3150 

9.9567 

9.9561 

9-9555 

9-9549 

9-9543 

9.6882 

0.3118 

9 - 95.37 

9.6914 

9.6946 
9.6977 

9.7009 
9.7040 

0.3086 

0.3054 

0.3023 

0.2991 

0.2960 

9-9530 

9.9524 

9.9518 

9.9512 

9 ■ 9505 

9.7072 

0.2928 

9.9499 

9-7103 

9-7134 

9-7165 

9.7196 
9.7226 

0.2897 
0.2866 
0.2835 
0.2804 
0.2774 

9.9492 

9.9486 

9-9479 

9-9473 

9.9466 

9-7257 

0.2743 

0-9459 

9.7287 

9-7317 

9-7348 

9-7378 

9.7408 

0.2713 

0.2683 

0.2652 

0.2622 

0.2592 

9-9453 

9.9446 

9-9439 

9-9432 

9.9425 

9-7438 

0.2562 

9.9418 

9.7467 

9-7497 

9.7526 

9-7556 

9-7585 

0.2533 

0.2503 

0.2474 

0.2444 

0.2415 

9 - 94 ri 
q .9404 

9-9397 

9.9390 

9-0383 

9.7614 

0.2386 

9-9375 

L.Cotg. 

d. c. 

L.Taruf 

L. Sin 


d. 


5 

4 

5 
5 
5 

4 

5 
5 
5 
5 
5 

5 

5 

6 

5 

5 

5 

6 

5 

6 

5 

6 

5 

6 

5 

6 
6 
6 

5 

6 

6 

6 

6 

6 

6 

6 

7 

6 

6 

6 

7 

6 

7 

6 

7 

6 

7 

7 

6 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

8 


d. 


50 

40 

30 

20 

10 

o 09 

50 

40 

30 

20 

10 

o 68 

50 

40 

30 

20 

10 

o 67 

50 

40 

30 

20 

10 

o 66 

50 

40 

3 ° 

20 

10 

o 65 

50 

40 

30 

20 

10 

o 64 

50 

40 

30 

20 

10 

o 63 

50 

40 

30 

20 

TO 

o 63 

50 

40 

3 ° 

20 

10 

o 61 

50 

40 
30 
20 
'10 
o 60 


FUNCTIONS. 2 19 

r • 


FUNCTIONS. 


P P. 


40 39 38 o 

1 4.0 3.9 3.80.5 

2 8.0 7.8 7.6 1.0 

3 12.0 IT. 7 II.4 I.5 

4 16.0 15.6 15.2 2.0 

5 20.0 19.5 19.0 2.5 

6 24.0 23.4 22.8 3.0 

7 28.0 27.3 26.6 3.5 

8 32.0 31.2 30.4 4.0 

9 36.0 35.1 34.2 4.5 

37 36 35 

1 3-7 3-6 3-5 

2 7.4 7.2 7.0 

3 11.1 10.8 10.5 

4 14.8 14.4 14.0 

5 18.5 18.0 17.5 

6 22.2 21.6 21.0 

7 25.9 25.2 24.5 

8 29.6 28.8 28.0 

9 33-3 3 2 '4 3 i -5 

34 33 32 6 

1 3-4 3-3 3-20.6 

2 6.8 6.6 6.41.2 

3 10.2 9.9 9.61.8 

4 13.6 13.2 12.8 2.4 

5 17.0 16.5 16.0 3.0 

6 20.4 19.8 19.2 3.6 

7 23.8 23.1 22.4 4.2 

8 27.2 26.4 25.6 4.8 

9 30.6 29.7 28.8 5.4 


31 30 29 7 

1 3.1 3.0 2.90.7 

2 6.2 6.0 5.8 1.4 

3 9.3 9.0 8.7 2.1 

4 12.4 12.0 11.6 2.8 

5 15-5 iS-o 14 • 5 3-5 

6 18.6 18.0 174 4.2 

7 21.7 21.0 20.3 4.9 

8 24.8 24.0 23.2 5.6 

9 27.9 27.0 26.1 6.3 

28 2 7 26 8 

1 2.8 2.7 2.6 0.8 

2 5.6 5.4 5.2 1.6 

3 8.4 8.1 7.8 2.4 

4 11.2 10.8 10.4 3.2 

5 i 4 -o 13.5 13-° 4 -o 

6 16.8 16.2 15.6 4.8 

7 19.6 18.9 18.2 5.6 

8 22.4 21.6 20.8 6.4 

9 25.2 24.3 23.4 7.2 

25 24 23 22 

1 2.5 2.4 2.3 2.2 

2 5.0 4.8 4.6 4.4 

3 7.5 7.2 6.9 6.6 

410.0 9.6 9.2 8.8 

5 12.5 12.0 11.5 11.o 

6 15.0 14.4 13.8 13.2 
717.5 16.8 16.1 15.4 
8 20.0 19.2 18.4 17.6 
o 22.5 21.6 20.7 19.8 


P. P. 



















































































































































220 


TRAVERSE INSTRUMENTS AND METHODS 


Table VI. 

LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 


0 / 

L. Sin. 

d. 

30 0 

9.6990 

22 

IO 

9.7012 


20 

9-7033 

21 

3 ° 

9-7055 

22 

40 

9.7076 

21 

5 ° 

9 • 7097 


31 0 

9.7118 

21 

IO 

9 7*39 

21 

20 

9.7160 

21 

3 ° 

9.7181 

21 

40 

9.7201 

21 

20 

50 

9.7222 

32 0 

9.7242 


IO 

9.7262 


20 

9.7282 

20 

30 

9.7302 

20 

40 

9.7322 

50 

9-7342 


33 0 

9 - 736 i 

*9 

20 

IO 

9.7380 

20 

9.7400 

30 

9 - 74 T 9 

x 9 

40 

9-7438 

*9 

50 

34 0 

9-7457 

9.7476 

*9 

*9 

18 

IO 

9.7494 

20 

9 - 75 I 3 

T 9 

18 

30 

9-7531 

40 

9 - 755 o 

*9 

18 

18 

50 

9.7568 

35 0 

9.7586 

18 

IO 

9.7604 

18 

18 

20 

9.7622 

3 ° 

9.7640 

40 

9-7657 

*7 

18 

50 

9-7675 

36 0 

9.7692 

1 7 

18 

IO 

9.7710 

20 

9.7727 

J 7 

30 

9-7744 

1 7 

40 

9.7761 

*7 

50 

9.7778 

*7 

37 0 

9-7795 

*7 

16 

IO 

9.7811 

20 

9.7828 

T 7 

16 

30 

9.7844 

40 

9.7861 

1 7 

50 

9.7877 

16 

38 0 

9-7893 

*7 

IO 

9.7910 

l6 

20 

9.7926 

30 

9.7941 

*5 

16 

16 

16 

40 

9-7957 

5 ° 

9-7973 

39 0 

9.7989 

IO 

9.8004 

*5 

20 

9.8020 

16 

30 

9.8035 

15 

40 

9.8050 

15 

16 

50 

9.8066 

0 

© 

9.8081 

15 


L. Cos. 

d. 


L.Tang 

9.7614 

97644 

9-7673 

9-7701 

9 - 773 ° 

9-7759 

9.7788 

9.7816 

9-7845 

9 7873 

9.7902 

9 793 ° 

9-7958 

9.7986 
9.8014 
9.8042 
9.8070 
9.8097 

9.8125 

9 - 8 i 53 

9.8180 

9.8208 
9-8235 
9.8263 

9.8290 

9-8317 

9.8344 

9 - 837 1 

9.8398 

9.8425 

9-8452 

9.8479 

9.8506 

9-8533 

9-8559 

9 8586 

9 8613 

9.8639 

9.8666 

9.8692 
9.8718 
9-8745 

9.8771 

9.8797 

9.8824 

9.8850 

9.8876 

Q.8902 

9.8928 

9.8954 

9.8980 

9.9006 

9 • 9 ° 3 2 
9.9058 

9.9084 

9.9110 

9 - 9*35 

9.9161 

9.9187 

9.9212 

9-9238 

L.Cotg. 


d. c. 


3 -> 

29 

28 

29 

29 

29 

28 

29 

28 

29 
28 
28 

28 

28 

28 

28 

27 

28 

28 

27 

28 

27 

28 
27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

26 

27 
27 

26 

27 
26 

26 

27 
26 

26 

27 
26 
26 
26 
26 

26 

26 

26 

26 

26 

26 

26 

25 

26 
26 

25 

26 


d. c. 


L.Cotg. 

0.2386 

0.2356 

0.2327 

0.2299 

0.2270 

O 2241 

0.2212 

0.2184 
' 0.2155 
0.2127 
0.2098 
0.2070 

0.2042 

0.2014 

0.1986 
0.1958 
0.1930 

0.1903 

0.1875 

0.1847 

0.1820 

0.1792 
0.1765 

0.1737 

O. T7IO 

0.1683 
0.1656 
0.1629 
0.1602 

0.1575 

0.1548 

0.1521 

0.1494 

0.1467 

0.1441 

0.1414 

0.1387 

0.1361 

0.1334 

0.1308 
0.1282 
0.1255 

O.1229 

0.1203 

0.1176 
0.1150 
0.1124 
0.1098 

O.IO72 

O.IO46 
O.1020 

O.OO94 

O.O968 

O.O942 

0.0916 

0.0890 
0.0865 
0.0839 
0.0813 
0.0788 

0.0762 

L.Tang 


L. Cos. 

9-9375 

9.9368 

9.9361 

9 9353 
9.9346 

9-9338 

9 - 933 * 

9 9323 
9 - 93*5 
9.9308 

9 - 93 °° 

9.9292 

9.9284 

9.9276 

9.9268 

9.9260 

9.9252 

9.9244 

9.9236 

9.9228 
9.9219 
9.9211 
9.9203 
9.9194 

9.9186 

9.9177 

9.9169 

9.9160 

9 - 9 * 5 * 

9.9142 

9 - 9*34 

9.9125 

9.9116 

9 - 9 io 7 

9.9098 

9.9089 

9 9080 

9.9070 

9.9061 

9.9052 

9.9042 

9.9033 

9.9023 

9.9014 

9.9004 

9.8995 

9.8985 

9-8975 

9.8965 

9-8955 

9-8945 

9-8935 

9.8925 

9.8915 

9.8905 

9.8895 

9•8884 
9.8874 
9.8864 
9-8853 

9.8843 

L. Sin. 


d. 


9 

9 

9 

9 

9 

10 

9 

9 

10 

9 

10 

9 
10 

9 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

11 

10 

10 

11 


d. 



P. P. 

0 60 


22 

21 

30 

5 ° 

I 

2.2 

2.1 

3 -o 

40 

2 

4-4 

4.2 

6.0 

3 ° 

3 

6.6 

6-3 

9.0 

20 

4 

8.8 

8.4 

12.0 

IO 

5 

II .O 

10.5 

15.0 

0 50 

6 

13-2 

12.6 

18.0 

7 

* 5-4 

14.7 

21.0 

50. 

8 

17.6 

16.8 

24.0 

40 

9 

19.8 

18.9 

27.0 

3° 





20 





IO 


20 

29 

4 

0 58 

I 

2.0 

2.9 

0.7 


2 

4.0 

5-8 

1.4 

5 ° 

3 

6.0 

8.7 

2.1 

40 

4 

8.0 

ii .6 

2.8 

30 

5 

10.0 

* 4-5 

3-5 

20 

6 

12.0 

17.4 

4.2 

IO 

7 

14.0 

20.3 

4.9 

0 57 

8 

16.0 

23.2 

5.6 


9 

18.0 

26.1 

6-3 

5 ° 





4 ° 





3 ° 


10 

28 

8 

20 

I 

1.9 

2.8 

0.8 

IO 

2 

3-8 

5-6 

1.6 

0 56 

3 

5-7 

8.4 

2.4 

50 

4 

7.6 

II . 2 

3-2 

5 

9-5 

I4.O 

4.0 

4 ° 

6 

11.4 

16.8 

4.8 

3 ° 

7 

* 3-3 

19.6 

5-6 

IO 

8 

15.2 

22.4 

6.4 

9 

*7.1 

25.2 

7.2 

0 55 





5 ° 

40 


18 

2 7 

9 

30 

I 

1.8 

2.7 

0.9 

20 

2 

3-6 

5-4 

1.8 

IO 

3 

5-4 

8.1 

2.7 

0 54 

4 

7.2 

10.8 

3.6 


5 

9.0 

* 3-5 

4-5 

50 

6 

10.8 

16.2 

5-4 

40 

7 

12.6 

18.9 

6-3 

30 

8 

14.4 

21.6 

7.2 

20 

9 

16.2 

24-3 

8.1 

IO 





0 53 


17 

26 

10 

5 ° 

I 

*•7 

2.6 

I .O 

2 

3-4 

5-2 

2.0 

40 

3 

5 -* 

7.8 

3 -o 

3 ° 

4 

6.8 

10.4 

4.0 

20 

5 

8-5 

13.0 

5 -o 

IO 

6 

10.2 

15.6 

6.0 

o52 

7 

11.9 

18.2 

7.0 

50 

8 

13.6 

20.8 

8.0 

40 

9 

* 5-3 

23-4 

9.0 

3 ° 





20 


16 

25 

11 

IO 

I 

i .6 

2-5 

1.1 

o 51 

2 

3-2 

5-o 

2.2 

50 

3 

4.8 

7-5 

3-3 

4 

6.4 

10.0 

4-4 

40 

5 

8.0 

*2.5 

5-5 

3 ° 

6 

9.6 

15.0 

6.6 


7 

II . 2 

*7-5 

7-7 

IO 

8 

12.8 

20.0 

8.8 

0 

w 

© 

9 

14.4 

22.5 

9.9 

/ O 

P. P. 















































































































































MAGNETIC DECLINATION . 


221 


Table VL 

LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 


0 / 

L. Sin. 

d. 

L Tang 

0 

0 

9.8081 

^5 

T 5 

9.9238 

• 10 

9.8096 

9.9264 

20 

9.8m 

14 

9.9289 

30 

9.8125 

15 

9 - 93*5 

40 

9.8140 

15 

9-934 * 

50 

9-8155 

14 

0.0366 

41 0 

9 8169 

9 - 93 Q 2 

IO 

9.8184 

T 5 

14 

9 - 94*7 

20 

9.8198 

15 

9 9443 

30 

9.8213 

T 4 

9.9468 

40 

9.8227 

14 

9-9494 

50 

9 8241 

m 

9 - 95*9 

42 0 

9-8255 

9-9544 

IO 

9.8269 

M 

14 

9-9570 

20 

9.8283 

14 

9-9595 

30 

9.8297 

14 

9.9621 

40 

9-8311 

1 1 

Q.Q646 

50 

0-8324 

14 

9.0671 

43 0 

9.8338 

9.0607 

IO 

9-8351 

T 3 

14 

Q.Q 722 

20 

9-8365 

1 $ 

9-9747 

30 

9.8378 

13 

9-9772 

40 

9.8391 

14 

9.9798 

50 

9•8405 


0.9823 

44 0 

9.8418 

l 3 

9.9848 

10 

9.8431 

1 3 

13 

9.9874 

20, 

9-8444 

13 

9.9809 

30 

9-8457 

I 2 

9.9924 

40 

9.8469 

13 

9.9949 

50 

9.8482 

13 

9-9975 

45 o 

9-8495 

O.OOOO 

L. Cos. 

d. 

I Cotg. 


d. c. 


26 

25 

26 
26 

25 

26 

25 

26 

25 

26 

25 

25 

26 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 


d. c. 


L.Cotg. 

L. Cos. 

0 0762 

9 8843 

0.0736 

0 0711 
0.0685 
0.0659 
0.0634 

9 8832 
9.8821 

9 8810 
9.8800 
Q.8789 

0.0608 

9-8778 

0.0583 

0.0557 

0.0532 

0.0506 
0.0481 

9.8767 

9 8756 
9-8745 
9-8733 
0.8722 

0.04-6 

9.8711 

0.0430 

0.0405 
0.0379 

0 0354 
0.0329 

9.8699 
9.8688 
9.8676 
9.8665 
9865 ? 

0.0303 

9.8641 

0.0278 

0.0253 

0.0228 

0.0202 

0.0177 

9.8628 

9.8618 

9.8606 
9.8594 
n.8582 

0.01^2 

9 85 6 9 

O.OT 26 

O.OIOI 

O.OO76 

O .OO5 I 
0.0025 

9-8557 

9-8545 

9 8532 
9.8520 
9.8507 

0.0000 

9-8495 

L.Tang 

L. Sin. 


d. 


11 
11 
11 

10 

11 

11 
11 
11 

11 

12 
11 

11 

12 

11 

12 
I I 
12 
12 

12 
I I 
12 
12 
12 

13 
12 
12 

*3 

12 

*3 

12 


30 

20 

IO 


48 


50 

40 

30 

20 

TO 


47 


50 

40 

30 

20 

TO 

o 46 

50 

40 

30 

20 

10 

o 45 


d. 



P. P. 

0 50 


15 

14 

13 

12 

50 

1 

*•5 

1.4 

*•3 

1.2 

40 

2 

3 -o 

2.8 

2.6 

2.4 

3 ° 

3 

4-5 

4.2 

3 9 

3-6 

20 

4 

6.0 

5-6 

5-2 

4*8 

IO 

5 

75 

7.0 

6-5 

6.0 

0 40 

6 

9.0 

8.4 

7.8 

7.2 


7 

10.5 

0.8 

9.1 

8.4 

50 

8 

12.0 

11 .2 

IO.4 

9.6 

40 - 

9 

* 3-5 

12.6 

11.7 

10.8 


P. P. 


in the direction of the sight the graduation on the circle 
may be read. 

In running a traverse with prismatic compass, distances 
are determined by pacing, timing an animal or a boat (Arts. 
95 and 96), or by other exploratory method, and the 
record of the distance and of the angle read for the course 
run are entered on the same line in the note-book. These 
quantities can be platted with scale and protractor, and give 
a fair plan of the route traveled. 

92. Magnetic Declination. —The compass-needle points 
to two magnetic poles, which coincide with the true north 
and south in but few places on the surface of the earth. The 
angle made with the true meridian by the magnetic meridian 
at any point is called the magnetic declination. Declination 

































































































222 


TRAVERSE INSTRUMENTS AND METHODS. 


is subject in all places to changes which are diurnal, secular, 
annual, and lunar. The two latter are very small and may 
be neglected. 

The diurnal variation is scarcely perceptible in any 
ordinary survey, being zero between io and I I in the morning 
and at about 8 P.M. It is greatest, that is, the north end is 
farthest east at about 8 in the morning, and farthest west at 
about i 130 in the afternoon. The limits of this diurnal 
variation are from five to fifteen minutes. The secular 
variation is quite important; it is fairly periodic in character 
and takes from 250 to 400 years to make a complete cycle. 

Declination may be determined at any time by an obser¬ 
vation on Polaris to ascertain the true north and its compari¬ 
son with the magnetic north (Chap. XXXIII). 

93. Secular Variation and Annual Change. —Owing to 
secular variation the declination determined at any date, say 
when some old survey was executed, has varied since. 
Therefore, compass-readings recorded on any date for a 
particular line will not agree with those observed for the 
same line or direction at another time. It is, accordingly, 
difficult to rerun old compass lines, and this can only be done 
with any degree of approximation by knowing the declination 
of the place for the date of the survey and reducing it to 
the present time. Numerous observations have been made 
by the various individual and government surveys, and from 
them there have been prepared diagrams and tables which 
aid in the determination of the declination at any known 
time. A line drawn on a map connecting points having the 
same magnetic declination is called an isogonic line, and the 
line joining points of no declination is called an agonic line. 

Plate IV, prepared from charts in the U. S. Coast and 
Geodetic Survey Report of 1905, shows the isogonic and 
agonic lines in the United States for the epoch of January, 
1905. The isogonic curves, which are lines of equal magnetic 
declination—that is, compass variation from true meridian,— 


LOCAL AT TRA CTION. 


223 


are shown for each degree. The plus sign indicates west, 
and the minus sign east declination. 

On the same plate are indicated in figures the amount of 
annual change of magnetic declination for the period 1905- 
1910. This change, or secular variation , indicates that the 
isogonic lines, as shown on the plate, are all moving west¬ 
ward, but not at the same rate; the movement being such 
that all western declinations are increasing and all eastern de¬ 
clinations are decreasing. Thus, to find the isogonic line for 
any year subsequent to 1905, the annual change which is indi¬ 
cated in minutes is to be applied, the plus sign signifying 
increasing west or decreasing east declination, and the minus 
sign the reverse. 

94. Local Attraction. —In running any survey, be it 
traverse or otherwise, by means of the compass-needle, the 
indications of the same are apt to be misleading as a result 
of local magnetic attraction. This is due to the needle being 
drawn from its mean position in any locality by the attraction 
of masses of magnetic iron-ore or of iron. In fact, if the 
compass is set up alongside of the tracks of a railroad or near 
the wheels of an iron-tired conveyance, it may be attracted 
from its normal position. It not uncommonly occurs that a 
closed traverse circuit run with a compass-needle will fail to 
check by a large error due to some such cause. 

Too much reliance cannot, therefore, be placed on com¬ 
pass traverses; and when there appears to be local attraction 
as shown by inaccurate closures of the surveys or of the lines 
run, the same must be allowed for in any subsequent work 
in the same locality. This is done by occupying every point 
or station in the traverse, or by reading backsights or bear¬ 
ings as well as foresights. It may be that local attraction 
will be so great in amount as to render it impossible to use 
the compass at all. During the running of any traverse with 
a compass, it is well to take the precaution of setting up 
and observing backsights and foresights on occasional lines, 
to determine whether local attraction exists. 


CHAPTER XI 


LINEAR MEASUREMENT OF DISTANCES. 

95. Methods of Measuring Distances; Pacing. —The 
most difficult element in running rough traverse or route sur¬ 
veys is the determination of distances. For directions either 
a cavalry sketch-board, a traverse plane-table with tripod, 
or a prismatic compass (Arts. 64, 61, and 91) may be em¬ 
ployed, while heights may be determined with the aneroid 
or by vertical angulation (Arts. 160 and 174). 

For the rough determination of distance, four methods 
may be employed, viz.: 

1. Measurement by odometer; 

2. By counting the paces of a man or animal; 

3. By use of the range-finder; or 

4. By time estimates. 

Where distances are to be measured with greater accuracy 
some of the following methods may be used, viz.: 

1. Tachymetric processes; 

2. Chains or steel tape; or 

3. Trigonometric processes. 

Where walking is necessary in order to get over the 
ground, very satisfactory and economic measures of distance 
can be had by pacing. With a little practice a degree of 
accuracy may be attained quite equal to that had in the 
direction and elevation measurements with the crude instru¬ 
ments employed -in reconnaissance work. It is desirable in 
pacing to adopt a stride shorter than the natural one; thus a 
man whose natural step in walking comfortably on level 

224 





























































































































DISTANCES BY FACING AND TIME. 


225 


ground is a yard long should adopt in pacing a stride of 32 
inches; and a man whose natural stride is 30 inches should 
adopt a 28-inch pace. The best way to ascertain the length 
of stride is to measure off a distance of, say, 200 feet and 
pace this several times, finding how many paces are required 
to measure the distance. If 70 strides are taken in this dis¬ 
tance, for instance, the pacer should adopt a stride which 
will enable him to make the distance in 80 steps, and should 
practice it with sufficient frequency to enable him to make 
the distance in 80 steps every time. Such a stride is practi¬ 
cally the normal one and is easy of calculation, since 40 paces 
equal 100 feet, 20 paces 50 feet, etc. Hence the number of 
paces multiplied by 2-J gives the distance in feet. With such 
a shortened stride the pacer can lengthen out a little when 
going up-hill, and shorten his stride in going down-hill, and 
he therefore should practice pacing not only on level ground 
but on inclined ground, to determine how to alter his stride. 

To further simplify pacing only every other step should 
be counted, as those of the left or right foot. In the 
foregoing case 20 double strides or the steps of one foot will 
equal 100 feet. Finally, as the length of this double stride 
is 5 feet, there will be nearly 1000 such steps to one mile; 
by lengthening the stride by practice to 5.3 feet, a thousand 
of these will almost exactly equal one mile of 5280 feet. 
Hence 100 such strides will measure one-tenth mile, etc. 

Excellent results have been obtained in rough geographic 
surveys by using instrumental measurements over portions of 
the country, and running checked cross-lines between by 
pacing. Numbers of such lines which the writer has had run 
and plotted checked out in distances of 10 or 15 miles 
between fixed points within | or of a mile, equivalent on a 
two-mile scale to or ^ of an inch. Such results have not 
only been obtained once, but day after day for years, and by 
different men, in the course of rough surveys over rugged 
mountains and deep gorges, through brush and fallen timber. 


226 


LINEAR MEASURE MEN 7' OF DISTANCES. 


Where it is possible to ride, fairly accurate results can be 
had by counting the paces of a saddle-animal. In deter¬ 
mining the pace methods somewhat similar to those used in 
determining human pacing should be employed, though of 
course no attempt can be made to shorten or create an 
artificial pace for the animal. Distances should be measured 
not only on level country but on hilly land, and these should 
be between a thousand feet and a quarter of a mile in len gth, 
and over these stretches the animal should be paced both at 
a walk and at a trot, until a fair average has been ascertained 
of the number of steps at each gait in traveling the distance, 
when the length of stride can be determined. It is a remark¬ 
able fact that the same animal exhibits great uniformity in 
the length of its stride under similar conditions. This is 
especially true of mules, which are the most satisfactory 
animals for use in pacing, as they are slower, steadier, and 
more uniform in their stride than horses. The writer has run 
many miles of traverse in the rough regions of the West and 
under varying topographic conditions, where the distances 
were measured by the pacing of an animal and checked in at 
either end by fixed locations, and the results were frequently 
as accurate as those obtained by average human pacing: this 
not only at a walk, but at a mixed gait, generally a moderate 
trot. In such manner as many as 30 to 35 miles of cross¬ 
country traverse have been run in a day, which were plotted 
on a geographic map on a scale of four miles to an inch with 
very satisfactory closure checks. In pacing with animals the 
stride of one fore foot only should be counted. 

96. Distances by Time.—Time estimates may be em¬ 
ployed where uniform pacing is impracticable. With little 
practice the horseman learns the rate of his animal , that is, 
the number of miles per hour which it traverses at different 
gaits, and in rough reconnaissances and exploratory work he 
is thus enabled to estimate with fair accuracy the distance 
he has traveled, by noting the time consumed in passing from 


MEASURING DISTANCES WITH LINEN TAPE. 22J 


one point to another, providing he pays close attention to 
the gaits of his animal and notes the time consumed with 
each different gait. 

In floating down a river a fairly satisfactory measure of 
the distance traveled can be obtained with currents of various 
velocities by timing floats over a measured distance in 
stretches of comparatively slow velocity, up to those in which 
the speediest rapids are encountered. The explorer may 
thus float down the stream, using a sketch-board or prismatic 
compass for direction, and by timing the boat from one 
course to another a fairly good survey may be made of the 
route traveled. Similar methods may be employed in ascer¬ 
taining the time necessary to row or paddle a boat in still 
water or against streams of varying velocities, and by en¬ 
deavoring to maintain a uniform rate in rowing or paddling 
it is possible by timing the courses to get a fair estimate of 
the distances. 

In platting paced and timed surveys it will be found desir¬ 
able to arrange a scale of pacing or timing. Thus, instead 
of transposing the number of paces into distance paced, a 
scale should be prepared on which should be graduated paces 
instead of distances (Fig. 55). For example, for a man who 
paces a yard at each stride, if the scale of plotting is to be one 
mile to the inch, there will be 10 paces to every ^ of an inch, 
and 100 paces to every ~ of an inch, so that by dividing 
an inch into 17.6 parts it will be equal to 100 paces, and 
lesser fractions can be interpolated. In the same manner, if 
a horse strides with the same foot a distance of 6 feet at each 
step, the inch may be divided into 88 parts, and each one of 
these will be equivalent to 10 strides. In similar manner a 
scale of time may be prepared, or, better still, in each case 
several scales for different strides or for different times. 
Thus, for a scale of one mile to one inch, 15 minutes’ travel 
at the rate of 3 miles an hour will be represented by f of an 


228 


LINEAR MEASUREMENT OF DISTANCES. 


inch, and the same time at the rate of 4 miles an hour will 
be represented by one inch. 

97. Measuring Distances with Linen Tape. —Various 
methods have been adopted for measuring distances on 
secondary and tertiary traverses in dense woods where the 
underbrush is so thick as to preclude the use of the stadia, 
and where the work required is such as to render unnecessary 
the accuracy attained by the use of steel tape or chain with 
two chainmen (Art. 99). Under such conditions two plans 
have generally been adopted: one, running of traverse lines 
by the topographer, directions being obtained by prismatic 
compass or plane-table (Arts. 91 and 61), and distances by 
the aid of an assistant who drags a chain; the other, by 
directions in the same manner, but distances by pacing (Art. 
95). As the topographer can see but a few yards ahead of 
him, he rarely sights to a fixed object, but on small-scale 
work finds it sufficient to sight in the direction in which the 
assistant has preceded him, dragging the chain. 

A more satisfactory and far more accurate mode of 
measurement under such circumstances has been found to 
consist in measuring distances with a long linen tape. This 
is made of tailor’s linen binding-tape obtained at dry-goods 
stores in spools of five hundred to one thousand feet in 
length, the best for this purpose being so finely woven and 
so smooth that it slips through the brush without catching, 
and is dragged ahead by one tapeman, the alignment of the 
tape giving the direction which the topographer is to sight 
for his azimuth. It is improved by immersion in boiling 
paraffin. The peculiarity of this apparatus consists in the 
fact that ordinarily the end of the tape will catch in brush 
and around trees, and tear and fray. To prevent this a 
narrow strip of celluloid, of the same dimensions as the tape, 
is sewed on its extreme end, the length of this celluloid 
appendage being from twelve to eighteen inches, and this 
causes it to slip between the bushes without becoming 


ODOMETER. 


229 


tangled or twisted. With such device numerous traverses 
have been run in the Adirondack woods and plotted on a scale 
of 1 ij inches to a mile, with average closure errors of to 
■j^Q- inch in circuits of 5 to 15 miles periphery. 

98. Odometer.—The odometer is not a distance-measurer, 
but a revolution-counter ; consequently a function of such dis¬ 
tance-measuring is the circumference of a wheel, the number 
of revolutions of which are counted. This wheel may be one 
of a buggy or other light conveyance, preferably a front 
wheel, in order that the odometer which is attached to it may 
be clearly in view at all times; or the wheel may be attached 
to a light hand-barrow, so that it can be trundled along trails 
or other routes over which two- or four-wheeled conveyances 
could not be driven. 

Distance-measuring by means of rolling a wheel over the 
surface and recording with the odometer the number of times 
the periphery of the wheel is applied to the surface may be 
done under the most favorable circumstances with nearly the 
accuracy of ordinary chain or stadia measuring. Such 
accuracy is not as great as that by the latter methods where 
they are carefully executed, but is sufficient for all purposes 
of distance-measuring where the results are to be plotted 
on a geographic map. 

The errors inherent in this work are of four kinds: 

1. Those due to the difficulty of reducing measures on an 
inclined surface to horizontal; 

2. Failure of the odometer or counter to correctly record 
the number of the revolutions; 

3. Slip or jolt in the wheel, due chiefly to striking 
stones, roots, and other obstacles; and 

4. Errors resulting from failure to run the wheel in a 
direct line between two station points. 

The first is perhaps the most serious, and as yet no satis¬ 
factory means have been devised whereby an instrument will 
record the changes in inclination passed over by an odometer 


230 


LINE A R MEASUREMENT 0E DISTANCES . 


wheel. The second may be partially guarded against only 
by using the best form of odometer and by the traverseman 
counting the revolutions of the wheel at the same time as a 
check. The third is not susceptible to correction, and errors 
due to this cause will occur unless the surface of the road be 
of exceptional quality. The errors due to the fourth cause 
may be practically eliminated by great care in driving or 
trundling the wheel in a straight line where the road surface 
will permit. These and like errors inherent in odometric 
surveys may be so greatly reduced by careful work as to 
render them of small moment when the survey is to be 
platted on a geographic map, and where there is sufficient 



Fig. 73.—Douglas Odometer Attached to Wheel. 


control by triangulation, stadia, or other equally good method 
to which to adjust the odometer traverses and thus eliminate 
their errors. As a general rule the errors due to odometer 
















































































ODOMETER AND HAND-RECORDER. 


231 


% 

measurement for this class of work are no greater than those 
introduced in the measurement of directions and due to the 
difficulty of plotting short road tangents to a small scale. 

There are several forms of odometer, among the best of 
which is the Douglas odometer, so named after its inventor, 
Mr. E. M. Douglas of the U. S. Geological Survey. (Fig. 
73.) This is firmly fixed to the axle of the wheel, and a cam 
is welded around the hub, the lift of the cam being of such 
height that as it strikes the lever of the odometer it raises 
this by just the amount sufficient to turn the cog-wheels 
within the instrument and move the index forward one 
division for each lift of the cam, corresponding to each revo¬ 
lution of the wheel. This odometer records revolutions 
directly, and a similar result may be obtained by the use of 
the ordinary printing-press counter, which may be suitably 
rigged on the axle of the wheel. The old form of pendulum 
odometer is so unreliable as to be of practically no value at 
all for purposes of surveying. 

Another form of odometer which has been found to be 
very satisfactory and accurate is the bell odometer (Fig. 74)* 



The record of this is in miles, tenths, and hundredths, instead 
of in number of revolutions. As a consequence it is manu¬ 
factured for different diameters of wheels. Knowing, there- 



232 


LINEAR MEASUREMENT OF DISTANCES. 


fore, the diameter of the wheel, the corresponding odometer 
must be ordered from the maker, and the record may be in 
miles or some scale unit, the latter being obtained by the 
surveyor making a false dial to be pasted over that which is 
furnished. This instrument is attached to the axle of the 
wheel and records by a small lug on the hub striking a star¬ 
shaped wheel connected with an endless screw within the 
odometer. 

The mode of counting revolutions of a wheel most satis¬ 
factory to expert traversemen is by tying a rag to one of the 
spokes and counting the revolutions as it comes in view each 
time. The traverseman soon becomes so expert that he does 
this counting without any apparent effort, and he intuitively 
catches through the corner of his eye the flash of the white 
cloth. Others fasten gongs with heavy pendulum clappers 
to the spokes of the wheel, so that each time the wheel 
revolves the clapper falls and strikes the bell. Others rig 
gongs to the axle and cause them to be struck by clappers 
attached to the revolving wheel or hub. The simplest 
counter is, however, the most certain, and of these is the cloth 
tied to the spoke and counted mentally, or, best of all, on a 
hand-recorder which is pressed at each flash of the cloth as it 



Fig. 75.—Hand Recorder. 


revolves, the recorder registering automatically the number 
of revolutions. (Fig. 750 

In using any form of odometer measurement it becomes 









Table VII. —for converting wheel revolutions into decimals of a mile. 

Prepared by S. S. Gannett. 


ODOMETER—REDUCTION TABLE . 


233 



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234 


LINEAR MEASUREMEN7' OF DISTANCES. 


necessary to accurately measure the outer circumference of 
the wheel and then, for convenience in plotting, to arrange a 
small table in accordance with the scale of the map, so that 
the number of revolutions multiplied into the circumference 
will give decimals of the map scale. This table would there¬ 
fore show for each tenth or hundredth of a mile or other 
unit the number of revolutions corresponding to such dis¬ 
tance. (Table VII.) 

99. Chaining. —The chain is little used by topographers 
excepting in dense woods and where the odometer cannot be 
run nor the stadia-rod seen. Chains are made of two 
lengths. The surveyor’s or Gunter s chain is 66 feet long 
and consists of 100 links, each 7.92 inches long. It is only 
used in determining the areas of land where the acre is the 
unit of measure. It is universally used in all of the United 
States public-land surveys. In all deeds of conveyance of 
property, where the word “chain” is used it refers to the 
66-foot chain. 

The engineer s cJiain ) that commonly used now, consists of 
100 links of steel wire, each connected with the next by two 
or three rings. Each link with its rings is one foot in length, 
and the total length of the chain is 100 feet. At intervals of 
ten links brass tags are brazed on, having one to four points 
to indicate distances of 10, 20, 30 feet, etc. The chain is 
provided at either end with brass handles fastened with a 
swivel, and the length of the chain includes the handles. 

The chain is done up from the middle, two links at a time 
being drawn into the hand. In measuring with the chain the 
front chainman carries a bundle of ten pins with pieces of 
cloth tied to them to attract the attention of the rear chain- 
man, and one of these pins is pressed into the ground at each 
chain-length. They are picked up and tallied by the rear 
chainman as he progresses. It is very easy to make mistakes 
in chaining, because of the liability to drop a chain in count¬ 
ing or the tendency to measure on sloping ground without 


CHAINING. 


235 


making the proper reductions to the horizontal. Moreover, 
the chain varies in length by tension, expansion, and contrac¬ 
tion. In chaining on steep ground the endeavor should be 
to hold the chain level and to plumb down from one end, and 
where the slopes are very steep half-chains of fifty feet only 
should be measured at a time in order that the chain may be 
held horizontally. 

The chainmen usually work ahead of the transitman 
towards the front flag, but they may be passed by him and 
follow him. The rear chainman is the more important of 
the two, as under ordinary methods of running he lines in the 
front chainman. The latter walks ahead, dragging the chain 
behind him, and moves to one side or the other according to 
the rear chainman’s signaled directions. The rear chainman 
shakes the chain out to get rid of kinked links, holds the end 
of the handle against his pin, and when the front chainman 
is in line calls out “ Down! ” when the latter places the fore 
pin in the ground. The front chainman should be the more 
active of the two, as the speed of the party depends upon his 
movements. 


CHAPTER XII. 


STADIA TACHYMETRY. 

100 . Tachymetry —Tachymetry, or, as it is sometimes 
called, tachyometry, is a method of rapid surveying which 
enables the operator, by a single observation upon a rod, to 
obtain the necessary horizontal and vertical data for the de¬ 
termination of the three elements of position of a point on 
the surface of the earth. Optical measurement of distances, 
azimuths, and heights by one observation is its essential prin¬ 
ciple, and is performed by means of stadia, telemeter, or special 
tachymetric attachment. Tachymetry furnishes at one oper¬ 
ation all the controlling elements required in topographic 
surveying as distinguished from plane surveying, which for 
topographic requirements must be supplemented by hypso¬ 
metric surveying. 

The instruments employed in tachymetric measurement 
consist of a good transit or plane-table and alidade (Arts. 85, 
56 and 59), and of a well-made rod variously divided accord¬ 
ing to the method employed (Art. 112). The requirements 
of this method are rapidity and comparative accuracy of meas¬ 
urement accomplished with the least cost, rather than with 
extreme precision. Where it is necessary to measure dis¬ 
tances and elevations at the same time, tachymetry gives as 
nearly accurate results, at much less cost of time and money, 
as are possible with chain and spirit-level. 

There are practically two systems of tachymetric meas¬ 
urement : 

1. The angular or tangential system ; and 

2. The stadia, telemeter, or subtend system. 

236 


TOPOGRAPHY WITH STADIA. 


23 7 


By angular tachymetry the horizontal distances are deter¬ 
mined by measuring the vertical angle between two marks at 
a given distance apart on a rod. By subtend or stadia 
tachymetry the horizontal distance is determined by observing 
the number of divisions intercepted on a rod between two 
lines in the diaphragm of the instrument, the distance between 
which bears a fixed ratio to the distance intercepted. 

The simplest form of tachymetry is with the plane-table, 
since on this is executed a graphic triangulation which attains 
the same end as does optical tachymetry or the range-finder 
(Art. 116). All forms of tachymetry are by means of trian¬ 
gulation, varying from a long base, as with plane-table or 
theodolite triangulation, to the short base of a Welden range¬ 
finder or of stadia-wires. Tachymeters may be divided into 
three classes : 

1. Those in which the measured base forms an integral 
part of the instrument itself, as is the case with the Wagner- 
Fennel type of tachymeter and the large fixed range-finders 
employed at seacoast batteries and on board ships; 

2. Those in which the measured base is on the point ob¬ 
served, as is the case with the stadia; and 

3. Those in which the base is measured on the ground at 
the observer’s station, as with the Welden range-finder and 
the plane-table used in range-finding. 

101. Topography with Stadia —In running a simple 
stadia traverse considerable topography may be obtained by 
the method of triangulation intersections (Art. 73) in con¬ 
junction with the stadia traverse, where it may be necessary 
to expand the area of topographic mapping. Thus signals 
may be established on commanding summits visible from the 
line, and directions and vertical angles (Arts. 54 and 160) 
be read to these, thus determining their positions. Then 
the stations marked by the signals can be occupied by the 
topographer; or else two or three assistants with stadia-rods 
may move about to the various positions which it is desired 


2 3 8 


5 TA DI A TA CH YME TR Y. 


to determine from these stations; the topographer meantime 
observing on the rods and thus locating them, after which he 
sketches in the topography adjacent to their position. 

The simplest method of surveying a river or narrow lake is 
with the stadia. The transit or plane-table should be carried 
in one boat and landings be made for stations; or, where 
banks will not permit of landing, the instrument may be firmly 
fastened in the boat. In a second boat a stadia-rod is carried 
and distances and directions are read to the rod by the topog¬ 
rapher, thus locating its position. By moving along, the 
topographer alternately passing the stadia-man and the stadia- 
man the topographer, and repeating this process, it is possible 
to procure a fair map of such river at moderate cost and in a 
comparatively short time. Greater speed and accuracy are 
obtainable, where the conditions permit, by executing plane- 
table triangulation in conjunction with the stadia measure¬ 
ments, according as one or other method is more convenient. 

102 . Tachymetry with Stadia. —The stadia is a device 
for determining the distance of a point from the observer by 
means of a graduated rod and the distance subtended on it 
by auxiliary wires in the telescope of a transit or alidade. 
The principle upon which stadia measurements are based is 
the geometric one that the lengths of parallel lines subtending 
an angle are proportioned to their distances from its apex. 
This proportion is applied through the medium of two fine 
wires or cross-hairs, or a glass with lines etched on it at the 
positions of the cross-hairs, and equidistant from the cen¬ 
tral cross-hair or line. The space which any two of these 
lines subtends on a rod or other object of known length 
bears a direct ratio to the distance of that object from the 
cross-hairs of the instrument, and, accordingly, knowing the 
distance subtended on the rod, its distance from the instru¬ 
ment can be at once determined. 

The term stadia surveying is used to include not only the 
measurement of the horizontal distance, but also the deter- 


TACHYMETRY WITH STADIA. 


239 


mination of heights by means of vertical angles observed to 
a fixed point on the rod. The stadia hairs may be horizontal 
and the rod held vertical, or vice versa , though the former 
method is usually preferred, for the rod can be more steadily 
and readily held in a vertical position than horizontally. The 
stadia-rod (Art. 112) may be held at right angles to the line 
of sight, which on a uniformly sloping hill would require it 
to be inclined at exactly right angles to the slope, or it may 
be held vertically, which is a much simpler operation, and 
the angle of inclination is then reduced by computation or 
tables (Arts. 104 and 105). The latter method is more 
safely and commonly employed than the former. 

The stadia-hairs are usually three in number and are 
placed parallel to each other, the outer equally distant from 
the center one, and at an extreme distance from each other 
which bears a decimal ratio between the distance subtended 
and that measured horizontally on the ground. For very 
accurate work it is considered better practice to have the 
stadia-hairs fixed so that they are not adjustable, and to de¬ 
termine experimentlly the ratio between the distance which 
they subtend on the rod and that measured on the ground ; 
or, in other words, the multiple of the distance subtended. 
The more common and convenient way, however, is to place 
the extreme hairs at such a distance apart that one foot sub¬ 
tended on the rod represents one hundred feet on the ground 
plus the focal length, and this ratio is obtained by having 
the hairs adjustable so that by testing the adjustment it can 
be ascertained at any time whether or not this ratio is cor¬ 
rectly fixed. By this means it is possible to measure greater 
distances than would be observable on a rod of given length, 
by using the half-hairs, or the distance between one of the 
extreme hairs and the middle hair; in which case a given dis¬ 
tance on the rod would correspond to double the distance on 
the ground measured by the extreme hairs. 

Example: One foot subtended by extreme hairs equals 


240 


5 TA DI A TA CH Y ME TR Y. 


ioo feet in distance; then, with level horizontal, if 5.68 feet 
are subtended on rod by hairs, the rod is distant from the 
telescope 568 feet plus the focal length f. 

Example: If the sight be still horizontal but the half 
hairs set so that 1 foot on the rod equals 200 feet in distance, 
then for the above intercept, 5.68 feet, the distance from the 
center of the instrument to the rod will be 1136 feet plus the 
focal distance. 

103. Accuracy and Speed of Stadia Tachymetry. —The 

accuracy and precision of well-conducted stadia-work is rarely 
fully appreciated. The stadia is essentially intended to secure 
rapidity rather than accuracy; nevertheless, with proper care 
to eliminate the chief sources of error, a high degree of accu¬ 
racy may be attained. It is now generally believed by most 
of those who have employed the stadia in careful operations 
that where properly handled it will produce results as good 
as, and frequently better than, those with the chain, especially 
in rough country where the inclination of the ground affects 
chaining most seriously. 

The degree of precision is dependent upon several quanti¬ 
ties, chief among which are: 

1. Length of sight; 

2. Ratio of the space subtended on the rod to the dis¬ 
tance on the ground ; 

3. Magnifying power of the telescope; 

4. Fineness of the cross-hairs; and 

5. Precautions taken to modify or eliminate the effects 
of refraction. 

Numerous experiments have been made to ascertain the 
effects of magnifying power . By observing distances with 
telescopes magnifying 15 times and 25 times respectively, 
under essentially the same conditions, Prof. Ira O. Baker 
found the average error in the first case, that is, with the 
lower magnifying power, to be 1 in 282, and in the second 
case, with the higher magnifying power, 1 in 333. He simi- 


ACCURACY AND SPEED OF STADIA. 


241 


larly experimented with a view r to determining the length of 
sight and corresponding error, with the result that the errors 
at distances of 100, 200 and 300 feet, respectively, were 1 in 
282, 1 in 263, and 1 in 370. These results, however, are not 
as valuable in showing the effect of this form of error, 
because it is largely introduced by the quality of the instru¬ 
ment, its magnifying power, size of cross-hairs, atmospheric 
conditions, and similar modifying circumstances. 

Experiments by Mr. R. E. Middleton showed the limit of 
accuracy of the stadia instrument with which he was experi¬ 
menting to be about 800 feet. Between 100 and 800 feet 
the average error was minus .43 feet per thousand feet; 
beyond this distance it increased rapidly to minus .97 feet 
per thousand. 

Perhaps some of the most interesting results obtained with 
stadia, as showing its precision , were those obtained by Mr. 
J. L. Van Ornum in taking topography on the international 
survey of the Mexican Boundary. The whole of the boundary 
line was measured with the stadia, and a large portion of it by 
the chain, and always tied in by a system of accurate primary 
triangulation. Corresponding distances were found by stadia 
and chain and compared with the known distances as obtained 
by triangulation, with the following results: 

Of five different stretches measured by the three methods, 
the total distance shown by triangulation was 99,110 meters, 
by stadia 99,025 meters, by corrected chain 99,041 meters. 
The total ratio of error between triangulation and chaining 
was minus I in 1436, and between triangulation and stadia 
minus I in 1166. Other sections of the line were measured 
by stadia and triangulation, but not by chain. In all there 
were measured 182.5 miles by stadia which were triangulated 
and in which the total difference in length was plus 50 
meters, or 1 in 5837. It may be noted that the chained 
distance was marked corrected chain, because in six measure¬ 
ments of the chained distance, dropping or omission of chain- 


242 


STADIA TA CH YMETR Y. 


lengths occurred which were detected in every instance by the 
stadia. The cause of this may be sought in the fact that the 
responsibility for correct measurement with stadia was placed 
on a trained instrumentman, thus reducing the danger of 
systematic error to a minimum. Moreover, there were some 
distances measured with the stadia which it would have been 
almost impossible to measure with the chain owing to the- 
roughness of the country and the great error and confusion 
which would have resulted from breaking chain frequently on 
exceedingly rough ground. 

In the surveys of the Indian Territory by the United 
States Geological Survey, a considerable number of section 
lines were run with stadia and chain with a view to determin¬ 
ing in a general way whether the stadia measurements were 
as accurate as the chain measurements, or, in other words, if 
they could be kept within the limit of error allowed in the 
Manual of the U. S. Land Office. This work was not done 
with any great accuracy, but with sufficient care to ascertain 
the fact that in every case the stadia measurements were 
well within the closing limits allowed for chaining according 
to the Manual, namely, f of a link per chain-length. 

The degree of accuracy which may be attained by com¬ 
paratively crude stadia surveys extending over a great area 
may be illustrated by a preliminary line run in Mexico by 
Mr. W. B. Landreth and the author from the west coast 
near Culiacan across the Sierra Madre to Durango and back 
by a different route. The method of running consisted in 
reading declination and distance with a transit to a stadia-rod 
held as foresight, the instrumentman leaving a small sapling 
with a piece of paper at his instrument position to serve as a 
rearsight. Vertical angles were read also in connection with 
this line. The total distance closed as a circuit 606 miles in 
length. It was computed by latitudes and departures which 
closed within 1100 feet, while the average cost of the survey 
was about $3.00 per linear mile. 


STADIA FORMULA WITH PERPENDICULAR SIGHT. 243 

The speed of stadia surveys is far greater than that of 
chaining where the surface of the ground is rough, since 
sights of one to two thousand feet length can be taken. 
Under similar circumstances the chain has to be laid down 
and stretched every hundred feet. Again, in a winding 
canyon or hilly road the instrument may have to be set up 
every hundred feet, perhaps ten times in 2000 feet. A single 
set-up and sight with the stadia may make the same measure 
and be not only speedier but far more accurate. The latter 
because one relatively crude measure will be less liable to 
error than ten separate measures of both angle and distance. 
Finally, the relative speed is further increased and the cost 
reduced correspondingly when elevations are to be obtained. 
For with stadia this is accomplished with the same men and 
instruments in conjunction with the plane survey. With 
chain survey, however, another party is necessary to deter¬ 
mine elevations by spirit-level. 

Over smooth country surveys may be made by stadia still 
more rapidly than by chaining or leveling if the rodman be 
mounted or ride a wheel, and the instrumentman ride or 
drive. On the Mexican survey above cited as many as 16 
miles a day were often made, including the determination of 
height and the sketching of topography. Under favorable 
circumstances 10 to 20 miles a day can be made as easily as 
5 to 8 miles walking with chain or level. 

Surveys by stadia traverse and plane-table intersections 
combined have been made of the shore line of large lakes, as 
Raquette Lake in the Adirondacks, at remarkably low cost 
and with great accuracy. This lake has a most intricate 
shore line and contains many islands, all heavily wooded. 
Yet its 45 miles of outline were mapped on a scale of 1J inches 
to a mile by one topographer and one stadiaman, both in 
boats, in 12 days. 

104. Stadia Formula with Perpendicular Sight. — In 

sighting the rod from the telescope, the stadia-wires appear 



244 


STADIA TA CH YME TR Y. 


to be projected upon the rod, thus intercepting a fixed dis¬ 
tance upon it. In fact there is formed at the position of the 
stadia-wires an image of the rod which the wires intercept, 
and at points which are the respective foci of the two points 
subtended on the rod. If the object-glass be considered a* 
simple bi-convex lens, then, by the principle of optics, the 
rays from any point of an object converge to a focus at a 
straight line which is a secondary axis connecting the point 
with its image and passing through the center of the lens. 
This point of intersection of the secondary axis is the optical 
center. It follows that the lines cC and bB , Fig. 76, drawn 



Fig. 76. —Stadia Measurement on Horizontal. 


from the stadia-wires, will intersect the object at points 
corresponding to those which the wires cut on the image of 
the rod. From this follows the proportion 


— — —; hence d — ~a, 
pi 1 



in which d — the distance of the rod from the center of the 

instrument; 

p = the distance of the stadia-wires from the center 
of the objective; 

a = the distance intercepted on the rod by the 
stadia-hairs; and 

i — the distance of the stadia-hairs apart. 

If p remained the same for all lengths of sight, then 4 








STADIA FORMULA WITH PERPENDICULAR SIGHT. 245 


would equal a constant, and d would be directly proportional 
to a. Unfortunately p varies with the length of the sight, 
and the relation between d and a is therefore variable,, 
Representing the principal focal length by the letter /, and 
applying the general formula for bi-convex lenses, that the 
sum of the reciprocals of the conjugate focal distances of the 
convex lens is equal to the reciprocal of the focal length of 
the lens, we have 


i i _ i 
P + d~T 



in which two different focal distances of image and object, 
p and p r , are approximately the same as p and d respectively. 
Substituting the above formula in (2) and transposing, we 

get 

d = ia -\- f. .(4) 

Since in the above ^and i remain constant for sights of all 
lengths, the factor by which a is multiplied is a constant, and 
d is equal to a constant multiplied by the length of a -f- f. 

/. 

The constant corresponding to — is usually designated by k y 

and accordingly the distance from the rod to the objective of 
the telescope is equal to a constant times the reading on the 
rod plus the focal length of the objective. 

To obtain the exact distance to the center of the instru¬ 
ment, it is further necessary to add the distance of the 
objective from the center of the instrument to f, which may 
be called c\ the final expression for the distance of the hori¬ 
zontal sight is then 

d = ka + f + c .(5) 


The approximate value of f, the focal length, may be 
obtained by focusing the telescope on a distant object and 
measuring the distance from the center of the object-glass to 



246 


STADIA TA CHYME TR Y. 


the cross-hairs. The value of c is not constant in most 
instruments, since the objective is moved in focusing for the 
different distances. It may be determined by focusing on 
an object a few hundred feet away and measuring the distance 
from the objective to the center of the instrument. 

The value of 4 * = k may be determined by driving a 

l 

tackhead in a stake and setting up the instrument over this. 
From this point measure two distances—one, say, of 100 
feet, and the other of 300 feet—and holding a rod at each, 
note the space intercepted on the rod at each point. Now, 
from the formulas, 


and 


d=L + (f+c) 


d' = 


■ • ( 6 ) 


-, a ' + (/ + C )l 


• • ( 7 ) 


in each of which the values of d and d' and a and a are 

, ■ / 

known, the values of — = k and f c may be deduced. 

105. Stadia Formula with Inclined Sight. —Formula (5) 
is based on the assumption that the visual ray is horizontal 
and the rod held vertical; that is, the line of sight is assumed 
to be perpendicular to the rod. This formula is inaccurate, 
however, for most stadia-work, because the sights are not 
taken on a level, but usually on a slope or inclination. 
Accordingly, d is not the horizontal distance fiom the 
instrument to the rod, but the inclined distance from the 
horizontal axis of the telescope to the point on the rod 
covered by the central visual hair. Formula (5) may be used 
with an inclined sight, provided the rod is held perpendicular 
to the central visual ray, and such perpendicularity may be 
obtained by a telescope, or, more simply, by means of a pair 


STADIA FORMULA WITH INCLINED SIGHT. 247 

of sights attached to the rod at right angles, or by a plumb- 
bob. 

The effort to procure perpendicularity of the rod involves 
several serious difficulties, among which are: (1) the difficulty 
of holding the rod steadily in this position; (2) the fact that 
it is not always possible for the rodman to see the telescope 
at a long distance or great angle; and (3) because the 
formulas for computing the horizontal and vertical co¬ 
ordinates are more simple wb^n the rod is held vertical than 
when it is held perpendicular to the line of sight. 

The same effect as is obtained by perpendicularity is 
obtained by holding the rod horizontally and having the cross¬ 
wires of the telescope placed vertically. There are some 
advantages in this method, because there is no likelihood of 
confusing either of the stadia-hairs with the central horizontal 
leveling-hair used in obtaining elevations. Though in some 
detailed work this method has been employed, the chief 
objection to it is the difficulty of holding the rod horizontal, 
it being usually necessary to support it upon trestles or some 
similar device. Neither of the above methods is generally 
accepted, as they are less simple of accomplishment and pro¬ 
duce no better results than the more usual method of holding 
the rod vertical. Hence they will not be further discussed. 

The rod may be held vertical with as great ease as may a 
leveling-rod by balancing it between the fingers or by having 
attached to it plumbing-levels. The formula for reduction 
to verticality is comparatively simple. Let a = angle of 
central visual ray with the horizon. This angle is measured 
by the central stadia-hair, either as a process in determining 
trigonometric levels or merely with the object of reducing 
the stadia distances. It is generally small and should be 
kept as small as possible to produce the best results. Let 
2 /3 = the visual angle subtended by the extreme cross-hairs 
on the rod, an angle which is always small, rarely exceeding 
one half of a degree. 




248 


6 - TA D/A 1 ’A CH YME TR Y. 


As CD (Fig. 77) is the actual distance subtended on the 
rod, and AB the distance which would be subtended if the 



Fig. 77.—Stadia Measurement on Slope. 


rod were held perpendicular to the line of sight, the relation 
between AB and CD is required, and by simple mathematical 
deduction we obtain as an expression for this 

AB — CD cos a .(8) 

Since AB is the distance subtended with the rod held per¬ 
pendicular to the line of sight, it is the value of a correspond¬ 
ing to the distance d in Formula (5), and it therefore becomes 

d — ka cos oc (f-\- c) .(9) 

Let d = the horizontal distance from the center of the 
instrument to the vertical foot of the rod, the actual distance 
which it is desired to measure. Then d' = d cos oc = actual 
distance from center of instrument to center of rod multiplied 
by cos a. Substituting in this the value of d from formula 
(9), we have 

d r = ka cos’ a + (/+ c) cos oc. . . . (10) 

We also have 

sin 2 oc 

h =(/-{- c) sin a -f- ak --—, . . . (11) 

in which h — the vertical distance or height of the object 
above the instrument. 









DETERMINING HORIZONTAL DISTANCES . 249 


With the aid of these two formulae the horizontal and 
vertical distances can be immediately calculated when reading 
on a vertical rod and when the angle of elevation is observed. 
From them numerous stadia tables have been calculated, 
the earlier and more important of which were those published 
by Messrs. J. A. Ockerson and Jared Teeple and those pub¬ 
lished by Mr. Arthur Winslow. 

106. Determining Horizontal Distances from Inclined 
Stadia Measures.—The following table (VIII), derived from 
the U. S. Coast and Geodetic Survey reports, is one of the 
most compact for use in reducing short stadia sights observed 
on slopes to their horizontal projections. 


Table VIII. 

REDUCTION OF INCLINED STADIA MEASURES TO 
HORIZONTAL DISTANCES. 


Horizontal projection of— 


inclination 
in degrees. 

10 feet. 

20 feet. 

30 feet. 

40 feet. 

50 feet. 

60 feet. 

70 feet. 

80 feet. 

90 feet. 

Deg. 

I 

9-997 

19-995 

29-993 

39 99 i 

49.988 

59.986 

69.984 

79.981 

89.979 

2 

9.99 

19.98 

29.97 

39-96 

49-95 

59-04 

69.94 

79.92 

89.91 

3 

9.98 

i 9 - 9 6 

29-93 

39 - 9 i 

49.88 

59.86 

69.84 

79.82 

89.80 

4 

9.96 

19.92 

29.88 

39-84 

49.80 

59 - 7 6 

69-72 

79.68 

89.64 

5 

9.94 

19.88 

29.81 

39-75 

49.69 

59-63 

69-57 

79-50 

89.44 

6 

9.91 

19.82 

29-73 

39-64 

49-56 

59-46 

69-37 

79.27 

89.20 

7 

9.88 

19.76 

29.64 

39-52 

49.40 

59.28 

69.16 

79.04 

88.91 

8 

9.84 

19.68 

29-53 

39-37 

49.21 

59.06 

68.90 

78.74 

88.58 

9 

9.80 

19.60 

29.40 

39.21 

49.01 

58.80 

68.61 

78.41 

88.21 

IO 

9-75 

19.51 

29.27 

39.02 

48 78 

5 8 -54 

68.29 

78.05 

87.79 

IX 

9.70 

19.41 

29.Il 

38.82 

48.52 

58.22 

67.93 

77.64 

87-34 

12 

9- 6 5 

19.30 

28.95 

38.60 

48.24 

57-90 

67-55 

77.20 

86 84 

13 

9.60 

19.20 

28.80 

38.40 

48.00 

57.60 

67.20 

76.80 

86.40 

14 

9-55 

ig. TO 

28.65 

38.20 

47-75 

57-30 

66.85 

76.40 

85.95 

15 

9-50 

I9.OO 

28.50 

38.00 

47-50 

57.00 

66.50 

76.00 

85.50 


Example : d = 160.20 and <x= 7° 00'. d — 98.70 and a = 4 0 00. 

i 100. 98.80 (90. 89.64 

160.2-] 60. 59- 2 8 98*7 \ 8. 7.968 

( 0.2. 0.1976 ( 0.7. 0.6972 


d ' — 158.2776 = 98.3052 

107. Horizontal Distances and Elevations from Stadia 
Readings. —Table IX was computed by Mr. Arthur Winslow 
and is reproduced here from J. B. Johnson s tl Surveying. 


































250 


STADIA TA CH YMETR Y. 


Table IX.* 

Horizontal Distances and Elevations from Stadia Readings. 



o° 

1 

0 

2 

}° 

3 

0 

Minutes. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff 

Hor. 

Diff. 


Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev 

o . . 

100.00 

0.00 

99-97 

I.74 

99.88 

3-49 

99-73 

5- 2 3 

2 . . 

44 

0.06 

44 

I.80 

99.87 

3-55 

99.72 

5.28 

4 • • 

a 

0.12 

44 

1.86 

44 

3.60 

99.71 

5-34 

6 . . 

a 

O.17 

99.96 

1.92 

44 

3.66 

44 

5.40 

8 . . 

u 

O.23 

44 

1.98 

99.86 

3-7 2 

99.70 

5-46 

IO . . 

a 

O.29 

44 

2.04 

44 

OJ 

-vi 

00 

99.69 

5-52 

12 . . 

u 

o -35 

44 

2.09 

99.85 

3-84 

44 

5-57 

14 . . 

u 

0.41 

99-95 

2.15 

44 

3 - 9 ° 

99.68 

5-6 3 

16 . . 

ii 

0.47 

44 

2.21 

99.84 

3-95 

44 

5-69 

18 . . 

44 

0.52 

44 

2.27 

44 

4.01 

99.67 

5-75 

20 . . 

a 

0.58 

44 

2.33 

99-83 

4.07 

99.66 

5.80 

22 . . 

n 

0.64 

99.94 

2.38 

44 

4 -i 3 

44 

5.86 

24 . . 

it 

0.70 

44 

2.44 

99.82 

4.18 

99-65 

5.92 

26 . . 

99.99 

0.76 

44 

2.50 

44 

4.24 

99.64 

5-98 

28 . . 

4i 

0.81 

99-93 

2.56 

99.81 

4 - 3 ° 

99-63 

6.04 

30 . . 

44 

0.87 

44 

2.62 

44 

4-36 

44 

6.09 

32 . . 

<4 

o -93 

44 

2.67 

99.80 

4.42 

99.62 

6.15 

34 • • 

44 

0.99 

44 

2.73 

44 

4.48 

44 

6.21 

36 . . 

44 

1.05 

99.92 

2.79 

99-79 

4-53 

99.61 

6.27 

38 . . 

44 

1.11 

44 

2.85 

44 

4-59 

99.60 

6-33 

40 . . 

44 

1.16 

44 

2.91 

99.78 

4.65 

99-59 

6.38 

42 . . 

44 

1.22 

99.91 

2.97 

44 

4.71 

44 

6.44 

44 • • 

99.98 

1.28 

44 

3.02 

99-77 

4.76 

99-58 

6.50 

46 . . 

44 

i -34 

99.90 

3.08 

44 

4.82 

99-57 

6.56 

48 . . 

44 

1.40 

44 

3- x 4 

99.76 

4.88 

99-56 

6.61 

50 . . 

44 

1.45 

44 

3.20 

44 

4.94 

44 

6.67 

52 . . 

44 

I - 5 I 

99.89 

3.26 

99-75 

4.99 

99-55 

6-73 

54 • • 

44 

T -57 

44 

3 - 3 1 

99-74 

5-05 

99-54 

6.78 

56 . . 

99-97 

1.63 

44 

3-37 

44 

5 - 11 

99-53 

6.84 

58 . . 

44 

1.69 

99.88 

3-43 

99-73 

5- J 7 

99-52 

6.90 

60 . . 

44 

1.74 

44 

3-49 

44 

5-23 

99 - 5 i 

6.96 

vo 

O 

II 

0-75 

0.01 

o -75 

0.02 

0-75 

0.03 

0-75 

0.05 

c = 1.00 

1.00 

0.01 

1.00 

0.03 

1.00 

0.04 

1.00 

0.06 

c = 1.25 

I 25 

0.02 

1.25 

0.03 

I.25 

0.05 

1.25 

0.08 

J_ 


* From “ Theory and Practice of Surveying,” by Prof. J. B. Johnson. New York: John 
Wiley & Sons. 





























































STADIA REDUCTION TABLES. 


251 


Table IX. 

Horizontal Distances and Elevations from Stadia Readings. 


— - -- 

Minutes. 

4 


5 ° 

6° 

7 ° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hoi. 

Diff. 


Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

0 . . 

99 - 5 1 

6.96 

99.24 

8.68 

98.91 

IO.40 

98.51 

12.10 

2 . . 

44 

7.02 

99-23 

8-74 

98.90 

IO.45 

98.50 

12.15 

4 • • 

99 - 5 ° 

7.07 

99.22 

8.80 

98.88 

IO.51 

98.48 

12.21 

6 . . 

99.49 

7-13 

99.21 

8.85 

98.87 

IO.57 

98.47 

12.26 

8 • • 

99.48 

7.19 

99.20 

8.91 

98.86 

10.62 

98.46 

12.32 

10 . . 

99-47 

7-25 

99.19 

8.97 

98.85 

10.68 

98.44 

12.38 

12 . . 

99.46 

7-30 

99.18 

9-03 

98.83 

10.74 

98-43 

12-43 

14 . . 

4 ( 

7-36 

99.17 

9.08 

98.82 

10.79 

98.41 

12.49 

16 . . 

99-45 

7.42 

99.16 

9.14 

98.81 

10.85 

98.40 

I2 -55 

18 . . 

99-44 

7.48 

99-15 

9.20 

98.80 

10.91 

98-39 

12.60 

20 . . 

99-43 

7-53 

99.14 

9-25 

98.78 

10.96 

98.37 

12.66 

22 . . 

99.42 

7-59 

99 -I 3 

9 - 3 1 

98.77 

11.02 

98.36 

12.72 

24 . . 

99.41 

7.65 

99.II 

9-37 

98.76 

11.08 

98-34 

12.77 

26 . . 

99.40 

7.71 

99.IO 

9-43 

98.74 

11.13 

98-33 

12.83 

28 . . 

99-39 

7.76 

99.09 

9.48 

98-73 

11.19 

98.31 

12.88 

30 . . 

99-38 

7.82 

99.08 

9-54 

98.72 

11.25 

98.29 

12.94 

32 . . 

99-38 

7.88 

99.07 

9.60 

98.71 

11.30 

98.28 

13.00 

34 • . 

99-37 

7-94 

99.06 

9.65 

98.69 

11.36 

98.27 

x 3-°5 

36 . . 

99-36 

7-99 

99-05 

9.71 

98.68 

11.42 

98.25 

13 11 

38 . . 

99-35 

8.05 

99.04 

9-77 

98.67 

11.47 

98.24 

I 3- I 7 

40 . . 

99-34 

8.11 

99-03 

9-83 

98.65 

ri -53 

98.22 

13.22 

42 . . 

99-33 

8.17 

99.OI 

9.88 

98.64 

11.59 

98.20 

13.28 

44 • • 

99-32 

8.22 

99.OO 

9-94 

98.63 

11.64 

98.19 

r 3-33 

46 . . 

99 - 3 1 

8.28 

98.99 

10.00 

98.61 

11.70 

98.17 

r 3-39 

48 . . 

99-30 

8-34 

98.98 

10.05 

98.60 

11.76 

98.16 

r 3-45 

50 . . 

99.29 

8.40 

98.97 

IO.II 

98.58 

11.81 J 

98.14 

x 3 - 5 ° 

52 . . 

99.28 

LO 

06 

98.96 

10.17 

98-57 

11.87 

98.13 

r 3-56 

54 • . 

99.27 

8.51 

98.94 

10.22 

98.56 

11 -93 

98.I I 

13.61 

56 . . 

99.26 

8-57 

98-93 

10.28 

98.54 

11.98 

98.IO 

x 3-67 

58 . . 

99.25 

8.63 

98.92 

10.34 

98.53 

12.04 

98.08 

1 3-73 

60 . . 

99.24 

8.68 

98.91 

10.40 

98.51 

12.10 

98.06 

13-78 

<\ 

11 

0 

Ln 

0.75 

0.06 

0-75 

0.07 

0-75 

0.08 

O.74 

0.10 

C ~ 1 .00 

1.00 

0.08 

O.99 

0.09 

0-99 

O.II > 

O.99 

0.13 

<r= I.25 

1.25 

0.10 

I.24 

O.II 

I.24 

0.14 

I.24 

0.16 


































































252 


STADIA TA CH YME TRY. 


Table IX. 

Horizontal Distances and Elevations from Stadia Readings. 


Minutes. 

8° 

9 ° 

io° 

11° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

o . . 

98.06 

I 3-78 

97-55 

1 5-45 

96.98 

17.IO 

96.36 

l 8.73 

2 . . 

98.05 

13.84 

97-53 

I 5 - 5 I 

96.96 

17.16 

96.34 

18.78 

4 • • 

98.03 

13.89 

97 - 5 2 

* 5-56 

96.94 

17.21 

96.32 

18.84 

6 . . 

98.OI 

13-95 

97 - 5 ° 

15.62 

96.92 

17.26 

96.29 

18.89 

8 . . 

98.OO 

I4.OI , 

97.48 

i 5- 6 7 

96.90 

I 7-32 

96.27 

18.95 

10 . . 

97.98 

14.06 

97.46 

15-73 

96.88 

17-37 

96.25 

19.OO 

12 . . 

97-97 

14.12 

97-44 

15.78 

96.86 

17-43 

96.23 

19.05 

14 . . 

97-95 

14.17 

97-43 

15.84 

96.84 

17.48 

96.21 

19.II 

16 . . 

97-93 

I 4-23 

97.41 

15.89 

96.82 

17-54 

96.18 

19.16 

iS . . 

97.92 

14.28 

97-39 

1 5-95 

96.80 

17-59 

96.16 

19.21 

20 . . 

97.90 

14-34 

97-37 

16.00 

96.78 

17-65 

96.14 

19.27 

22 . . 

97.88 

14.40 

97-35 

16.06 

96.76 

17.70 

96.12 

19.32 

24 . . 

97.87 

14.45 

97-33 

16.11 

96.74 

17.76 

96.09 

I 9-38 

26 . . 

97-85 

I 4 - 5 I 

97 - 3 i 

16.17 

96.72 

17.81 

96.O7 

19-43 

28 . . 

97-83 

14.56 

97.29 

16.22 

96.70 

17.86 

96.05 

19.48 

30 . . 

97.82 

14.62 

97.28 

16.28 

96.68 

17.92 

96.03 

* 9-54 

32 . . 

97.80 

14.67 

97.26 

*6 -33 

96.66 

17.97 

96.OO 

19-59 

34 • • 

97.78 

14-73 

97.24 

16.39 

96.64 

18.03 

95-98 

19.64 

36 . . 

97.76 

14.79 

97.22 

16.44 

96.62 

18.08 

95-96 

19.70 

38 • • 

97-75 

14.S4 

97.20 

16.50 

96.60 

18.14 

95-93 

1975 

40 . . 

97-73 

14.90 

97.18 

j 6-55 

96-57 

18.19 

95.91 

19.80 

42 . . 

97.71 

M -95 

97.16 

16.61 

96.55 

18.24 

95.89 

19.86 

44 • • 

97.69 

15.01 

97 -M 

16.66 

96-53 

18.30 

95.86 

19.91 

46 . . 

97.68 

15.06 

97.12 

16.72 

96.51 

18.35 

95-84 

19.96 

48 . . 

97.66 

1 5 * 1 2 

97.10 

16.77 

96.49 

18.41 

95.82 

20.02 

50 . . 

97.64 

I 5- I 7 

97.08 

16.83 

96.47 

18.46 

95-79 

20.07 

52 . . 

97.62 

x 5- 2 3 

97.06 

16.88 

96-45 

18.51 

95-77 

20.12 

54 • • 

97.61 

15.28 

97.04 

16.94 

96.42 

18.57 

95-75 

20.18 

56 . . 

97-59 

1 5-34 

97.02 

16.99 

96.40 

18.62 

95-72 

20.23 

58 . . 

97-57 

15.40 

97.00 

17-05 

96.38 

18.68 

95.70 

20.28 

60 . . 

97-55 

15-45 

96.98 

17.10 

96.36 

18.73 

95.68 

20.34 

c — °-75 

0.74 

0.11 

0.74 

0.12 

O.74 

0.14 

0-73 

0.15 

c — 1.00 

0.99 

0.15 

0.99 

0.16 

O.98 

0.18 

0.98 

0.20 

C — 1.25 

1.23 

0.18 

1.23 

0.21 

I.23 

0.23 

1.22 

0.25 



















































STADIA REDUCTION TABLES. 


253 


Table IX. 

Horizontal Distances and Elevations from Stadia Readings. 


Minutes. 

12° 

13 ° 

14 ® 

15 ° 

Hor. 

Diff 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff 



Dist 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O . . 

95.68 

20.34 

94.94 

21.92 

94.15 

23-47 

93-30 

25.OO 

2 . . 

95-65 

20.39 

94.91 

21.97 

94.12 

23-52 

93-27 

25.05 

4 • • 

95-63 

20.44 

94.89 

22.02 

94.09 

23-58 

93-24 

25.IO 

6 . . 

95.61 

20.50 

94.86 

22.08 

94.07 

23-63 

93.21 

25 -I 5 

8 . . 

95.58 

20.55 

94.84 

22.13 

94.04 

23.68 

93 - 1 8 

25.20 

10 . . 

9556 

20.60 

94.81 

22.l8 

94.OI 

23-73 

93.16 

25.25 

12 . . 

95-53 

20 66 

94-79 

22.23 

93-98 

23.78 

93- 1 3 

25-30 

14 • . 

95 - 5 1 

20.71 

94.76 

22.28 

93-95 

23-83 

93.10 

25-35 

16 . . 

95-49 

20.76 

94-73 

22-34 

93-93 

23.88 

93-07 

25.40 

18 . . 

95-46 

20.81 

94.71 

22.39 

93 - 9 ° 

23-93 

93-°4 

25-45 

20 . . 

95-44 

20.87 

94.68 

22.44 

93-87 

23-99 

93 - 01 

25.50 

22 , . 

95-41 

20.92 

94.66 

22.49 

93-84 

24.04 

92.98 

25-55 

24 . . 

95-39 

20.97 

94-63 

22-54 

93.81 

24.09 

92.95 

25.60 

26 . . 

95-36 

21.03 

94.60 

22.60 

93-79 

24.14 

92.92 

25.65 

28 . . 

95-34 

21.08 

94.58 

22.65 

93-76 

24.19 

92.89 

25.70 

30 . . 

95-32 

21.13 

94-55 

22.70 

93-73 

24.24 

92.86 

25-75 

32 . . 

95-29 

21.iS 

94-52 

22-75 

93-70 

24.29 

92.83 

25.80 

34 • • 

95-27 

21.24 

94 - 5 ° 

22.80 

93-67 

24-34 

92.80 

m 

00 

in 

Cl 

36 . . 

95.24 

21.29 

94-47 

22.85 

93-65 

24-39 

92.77 

25.90 

38 • • 

95.22 

21.34 

94.44 

22.9I 

93.62 

24.44 

92.74 

25-95 

40 . . 

95- 1 9 

21.39 

94.42 

22.96 

93-59 

24.49 

92.71 

26.00 

42 . . 

95- 1 7 

21.45 

94-39 

23.OI 

93-56 

24-55 

92.68 

26.05 

44 • • 

95- 1 4 

21.50 

94-36 

23.06 

93-53 

24.60 

92.65 

26.10 

46 . . 

95 - 12 

2 i -55 

94-34 

23.II 

93 - 5 ° 

24.65 

92.62 

26.15 

48 . . 

95-°9 

21.60 

94-31 

23.16 

93-47 

24.70 

92.59 

26.20 

50 . . 

95-07 

21.66 

94.28 

23.22 

93-45 

24-75 

92.56 

26.25 

52 • • 

95.04 

21.71 

94.26 

23.27 

93-42 

24.80 

92-53 

26.30 

54 • • 

95.02 

21.76 

94-23 

23-32 

93-39 

24.85 

92.49 

26.35 

56 . . 

94.99 

21.81 

94.20 

23-37 

93-36 

24.90 

92.46 

26.40 

OO 

94-97 

21.87 

94.17 

23.42 

93-33 

24-95 

92-43 

26.45 

60 . . 

94.94 

21.92 

94-15 

23-47 

93-30 

25.00 

92.40 

26.50 

' = 0.75 

0-73 

0.16 

o -73 

0.17 

0-73 

0.19 

0.72 

0.20 

c ~ 1.00 

0.98 

0.22 

0.97 

O.23 

0-97 

0.25 

0.96 

0.27 

C- 1.25 

1.22 

0.27 

1.21 

O.29 

1.21 

0.31 

1.20 

0.34 






















































254 


STADIA TA CH YMETR Y. 


Table IX. 


Horizontal Distances and Elevations from Stadia Readings. 




16 ° 

17 ° 

18 ° 

19 ° 

.Minutes. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

o 

• • 

92.40 

26.50 

9145 

27.96 

90-45 

29-39 

89.40 

30.78 

2 

« • 

92-37 

26-55 

91.42 

28.OI 

90.42 

29.44 

89.36 

30-83 

4 

• • 

92.34 

26.59 

91-39 

28.06 

90.38 

29.48 

89-33 

30.87 

6 

• • 

92.31 

26.64 

91-35 

28.IO 

90-35 

29-53 

89.29 

30.92 

8 

. • 

92.28 

26.69 

91.32 

28.15 

90.31 

29.58 

89.26 

30-97 

IO 

• • 

92.25 

26.74 

91.29 

28.20 

90.28 

29.62 

89.22 

31.01 

12 

• • 

92.22 

26.79 

91.26 

28.25 

90.24 

29.67 

89.18 

31.06 

14 

• • 

92.19 

26.84 

91.22 

28.30 

90.21 

29.72 

89.15 

31.IO 

16 

• • 

92.15 

26.89 

91.19 

28.34 

90.18 

29.76 

S9.II 

3 I-I 5 

18 

• • 

92.12 

26.94 

91.16 

28.39 

90.14 

29.81 

89.08 

3 *-i 9 

20 

• • 

92.09 

26.99 

91.12 

28.44 

90.11 

29.86 

89.04 

3 r -24 

22 

• • 

92.06 

27.04 

91.09 

28.49 

90.07 

29.90 

89.OO 

31.28 

24 


92.03 

27.09 

91.06 

28.54 

90.04 

29.95 

88.96 

31-33 

26 

• • 

92.00 

27 .I 3 

91.02 

28.58 

90.00 

30.00 

88.93 

3 t -38 

28 

• • 

91.97 

27.18 

90.99 

28.63 

89.97 

30.04 

88.89 

31.42 

30 

• • 

91-93 

27.23 

90.96 

28.68 

89-93 

30.09 

88.86 

3 M 7 

32 

. 

91.90 

27.28 

90.92 

28.73 

89.90 

30.14 

88.82 

3i-5i 

34 

• • 

9 i -87 

27-33 

90.89 

28.77 

89.86 

30.19 

88.78 

3 I -56 

36 

• 

9L84 

27.38 

90.86 

28.82 

89.83 

30.23 

88.75 

31.60 

33 

• • 

91.81 

2743 

90.82 

28.87 

89.79 

30.28 

88.71 

3^-65 

40 

• • 

9 x -77 

27.48 

90.79 

28.92 

89.76 

30-32 

88.67 

31.69 

42 


9 T 74 

27.52 

90.76 

28.96 

89.72 

30 -37 

88.64 

3 x -74 

44 

• • 

9 I - 7 I 

27.57 

90.72 

29.OI 

89.69 

30.41 

88.60 

3 x -78 

46 


91.68 

27.62 

90.69 

29.06 

89.65 

30.46 

88.56 

3 x -83 

48 

• • 

91.65 

27.67 

90.66 

29.I I 

89.61 

30-51 

88-53 

3 x -87 

5 ° 

« • 

91.61 

27.72 

90.62 

29.15 

89.58 

30-55 

88.49- 

31.92 

52 

• • 

91.58 

27.77 

90-59 

29.20 

89-54 

30.60 

88.45 

31.96 

54 

• • 

9 i -55 

27.81 

90.55 

29.25 

89.51 

30-65 

88.41 

32.01 

56 

• • 

9 *- 5 2 

27.S6 

90.52 

29.30 

89.47 

30.69 

88.38 

32.05 

58 

• • 

91.48 

27.91 

90.48 

29-34 

89.44 

30-74 

88.34 

32.09 

60 

• • 

9 i -45 

27.96 

90.45 

29-39 

89.40 

30.78 

88.30 

32.14 

'= 

075 

0.72 

0.21 

O.72 

O.23 

0.71 

0.24 

0.71 

0.25 

c = 

1. 00 

0.86 

0.28 

0-95 

O.30 

0.95 

0.32 

o -94 

o -33 

c — 

1.25 

1.20 

°-35 

1.19 

O.38 

1.19 

0.40 

1.18 

0.42 






















































STADIA REDUCTION TABLES . 


255 


Table IX. 

Horizontal Distances and Elevations from Stadia Readings. 


Minutes. 

1 C 

O 

0 

21 ° 

22 ° 

23 ° 



/ 







Hot. 

Diff 

Hor. 1 

Diff. 

Hor. 

Diff 

Hor. 

Diff. 


Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

0 . . 

88.30 

3 2 - T 4 

87.16 

33-46 

85-97 

34-73 

S 4-73 

35-97 

2 . . 

88.26 

3 2 - lS 

87.12 

33 - 5 o 

85-93 

34-77 

84.69 

36.01 

4 • • 

88.23 

3 2 - 2 3 

87.08 

33-54 

85.89 * 

34.82 

84.65 

36-05 

6 . . 

88.19 

32.27 

87.04 

33-59 

85.85 

34.86 

84.61 

36.09 

8 . . 

88.15 

3 2 - 3 2 

87.OO 

33-63 

85.80 

34-90 

84.57 

36-13 

10 . . 

88.11 

32-36 

86.96 

33-67 

85.76 

34-94 

84.52 

36-17 

12 . . 

88.08 

32.41 

86.92 

3 . 3-72 

00 

C/i 

Is) 

34-98 

84.48 

36.21 

14 . . 

88.04 

• 32-45 

86.88 

33-76 

85.68 

35.02 

84.44 

36-25 

l6 . • 

88.00 

32-49 

86.84 

33 - 8 o 

85.64 

35-07 

84.40 

36.29 

18 . • 

87.96 

32-54 

86.80 

33-84 

85.60 

35 - 11 

84-35 

36-33 

20 • • 

87-93 

32.58 

86.77 

33-89 

85.56 

35-15 

84.31 

36-37 

22 . . 

87.89 

32-63 

86.73 

33-93 

85.52 

35*9 

84.27 

36.41 

24 • • 

87.85 

32.67 

86.69 

33-97 

85.48 

35-23 

84.23 

3645 

26 . • 

87.81 

32-72 

86.65 

34.01 

85.44 

35-27 

84.18 

3649 

28 . . 

8777 

32.76 

86.61 

34.06 

85.40 

35-31 

84.14 

36-53 

30 . . 

87.74 

32.80 

86.57 

34.10 

85-36 

35-36 

84.IO 

36-57 

32 . . 

87.70 

32.85 

86.53 

34-14 

8 S- 3 1 

35-40 

84.06 

36.61 

34 • • 

87.66 

32.89 

86.49 

34 -i 8 

85.27 

35-44 

84.OI 

36-65 

36 . . 

87.62 

32-93 

86.45 

34-23 

85.23 

35-48 

83-97 

36.69 

38 . . 

87.58 

32.98 

86.41 

34-27 

85-19 

35-52 

83-93 

36.73 

40 . . 

87-54 

33-02 

86.37 

34-31 

85 -I 5 

35-56 

83.89 

36.77 

42 . . 

87-51 

33-07 

86.33 

34-35 

85.11 

35.60 

83.84 

36.80 

44 • • 

87.47 

33 -n 

86.29 

34-40 

85-07 

35-64 

83.80 

36.84 

46 . . 

87-43 

33-15 

86.25 

3444 

85.02 

35.68 

83.76 

36.88 

48 . . 

87-39 

33-20 

86.21 

34-48 

84.98 

3572 

83.72 

36.92 

50 . . 

87-35 

33-24 

86.17 

34-52 

84.94 

3576 

83.67 

36.96 

52 . . 

87-31 

33-28 

86.13 

34-57 

84.90 

35.80 

83.63 

37.00 

54 • • 

87.27 

33-33 

86.09 

34.61 

84.86 

35-85 

83-59 

37-04 

56 . . 

87.24 

33-37 

86.05 

34-65 

84.82 

35-89 

83-54 

37.08 

58 . . 

87.20 

33-41 

86.01 

34-69 

84.77 

35-93 

83-50 

37-12 

60 . . 

87.16 

33-46 

85-97 

34-73 

84-73 

35-97 

83.46 

y.16 

II 

p 

tn 

0.70 

0.26 

0.70 

0.27 

0.69 

0.29 

O.69 

0.30 

c = 1.90 

0.94 

o -35 

0-93 

0-37 

0.92 

0.38 

O.92 

0.40 

'= 1.25 

i-«7 

0.44 

1.16 

0.46 

1.15 

0.48 

1.15 

0.50 

_ 






























































256 


STADIA TA CH YME TR Y. 


Table IX. 

Horizontal Distances and Elevations from Stadia Readings. 


Minutes. 

240 

25° 

26° 

27° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 



Dist 

Elev. 

Dist. 

Elev 

Dist. 

Elev. 

Dist. 

Elev. 

O . . 

83.46 

37.16 

82.14 

38.30 

80.78 

39-40 

79-39 

40.45 

2 . . 

83.41 

37.20 

82.09 

38.34 

80.74 

39-44 

79-34 

40.49 

4 • • 

§3-37 

37-23 

82.05 

38.38 

80.69 

39-47 

79-30 

40.52 

6 . . 

83-33 

37-2 7 

82.OI 

38.41 

80.65 

39-51 

79-25 

40.55 

8 . . 

83.28 

37-3 1 

81.96 

3845 

80.60 

39-54 

79.20 

40.59 

10 . . 

83.24 

37-35 

81.92 

3849 

80.55 

39-58 

79-15 

40.62 

12 . . 

83.20 

37-39 

81.87 

38.53 

80.51 

39.61 

79.11 

40.66 

14 . . 

83- 1 5 

37-43 

81.83, 

38.56 

80.46 

39-65 

79.06 

40.69 

16 . . 

83.11 

37-47 

81.78 

38.60 

80.41 

39-69 

79.01 

40.72 

18 . . 

83.07 

37-5 1 

81.74 

38.64 

80.37 

39-72 

78.96 

40.76 

20 . . 

83.02 

37-54 

81.69 

38.67 

80.32 

39-76 

78.92 

40.79 

22 . . 

82.98 

37.58 

81.65 

38.7I 

80.28 

39-79 

00 

06 

40.82 

24 . . 

82.93 

37.62 

8l.6o 

3875 

80.23 

39-83 

Cl 

00 

06 

40.86 

26 . . 

82.89 

37.66 

81.56 

38.78 

80.18 

39.86 

78.77 

40.89 

28 . . 

82.85 

37-7 0 

81.51 

38.62 

80.14 

39-90 

78.73 

40.92 

30 . . 

82.80 

37-74 

81.47 

38.86 

80.09 

39-93 

78.68 

40.96 

32 . . 

82.76 

37-77 

81.42 

38.89 

80.04 

39-97 

78.63 

40.99 

34 • • 

82.72 

37 - 8 i 

81.38 

38.93 

80.OO 

40.00 

78.58 

41.02 

36 . . 

82.67 

37-85 

8 1-33 

38.97 

79-95 

40.04 

78.54 

41.06 

3 S • • 

82.63 

37-89 

81.28 

39-00 

79.90 

40.07 

78.49 

41.09 

40 . . 

82.58 

37-93 

81.24 

39-04 

79.86 

40.11 

78.44 

41.12 

42 . . 

82.54 

37 - 96 

81.19 

39.08 

7981 

40.14 

78.39 

41.16 

44 • • 

82.49 

38.00 

81.15 

39 -11 

79 76 

40.18 

78.34 

41.19 

46 . . 

82.45 

38.04 

81.10 

39 1 5 

79.72 

40.21 

78.30 

41.22 

48 . . 

82.41 

38.08 

81.06 

39 -i 8 

79.67 

40.24 

78.25 

41.26 

50 . . 

82.36 

38-11 

81.01 

39.22 

79.62 

40.28 

78.20 

41.29 

52 . . 

82.32 

38.15 

80.97 

39.26 

79-58 

40.31 

78.15 

41.32 

54 • • 

82.27 

38.19 

80.92 

39-29 

79-53 

40.35 

78.10 

41-35 

56 . • 

82.23 

38.23 

80.87 

39-33 

79.48 

40.38 

78.06 

41.39 

58 . . 

82.18 

38.26 

80.83 

39-36 

79-44 

40.42 

78.01 

41.42 

60 . . 

82.14 

38-30 

80.78 

39-40 

79-39 

40.45 

77.96 

41.45 

^ = 0.75 

0.68 

0.31 

0.68 

0.32 

0.67 

0-33 

0.66 

o -35 

c = 1.00 

0.91 

0.41 

0.90 

043 

0.89 

0-45 

0.89 

. 0.46 

<•=1.25 

1.14 

0.52 

1!3 

0-54 

1.12 

0.56 

1.11 

0.58 


























































STADIA REDUCTION TABLES . 


257 


Table IX. 

Horizontal Distances and Elevations from Stadia Readings. 


Minnteg. 

28 ° 

29 ° 

30 ° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 


Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O . . 

77.96 

41.45 

76.50 

42.40 

75.OO 

43-30 

2 . . 

77 - 9 1 

41.48 

76.45 

4243 

74-95 

43-33 

4 • • 

77.86 

41.52 

76.40 

42.46 

74.90 

43-36 

6 . . 

I—» 

00 

1^. 

41-55 

76.35 

42.49 

74.85 

43-39 

8 . . 

77-77 

41.58 

76.30 

42.53 

74.80 

43-42 

10 . . 

77.72 

41.61 

76.25 

42.56 

74-75 

43-45 

12 . . 

77.67 

41.65 

76.20 

42.59 

74.70 

43-47 

14 . . 

77.62 

41.68 

76.15 

42.62 

74.65 

43-50 

16 . . 

77-57 

4 I - 7 I 

76.IO 

42.65 

74.60 

43-53 

18 . . 

77 - 5 2 

41.74 

76.05 

42.68 

74-55 

43-56 

20 . . 

77.48 

41-77 

76.OO 

42.71 

74-49 

43-59 

22 . . 

77.42 

41.81 

75-95 

42.74 

74-44 

43.62 

24 . . 

77-38 

41.84 

75 - 9 ° 

4277 

74-39 

43-65 

26 . . 

77-33 

41.87 

75-85 

42.80 

74-34 

43-67 

28 . . 

77.28 

41.90 

75.80 

42.83 

74.29 

43-70 

30 . . 

77-23 

4 i -93 

75-75 

42.86 

74.24 

43-73 

32 . . 

77 -i 8 

41-97 

75-70 

42.89 

74.19 

43-76 

34 • • 

77-13 

42.00 

75-65 

42.92 

74.14 

43-79 

36 . . 

77.09 

42.03 

75.60 

42.95 

74.09 

43.82 

38 . . 

77.04 

42.06 

75-55 

42.98 

74.04 

43-84 

40 . . 

76.99 

42.09 

75-50 

43 - QI 

73-99 

43-87 

42 . . 

76.94 

42.12 

75-45 

43-°4 

73-93 

43-90 

44 • • 

76.89 

42.15 

75-40 

43-°7 

73.88 

43-93 

46 . . 

76.84 

42.19 

75-35 

43.10 

73-83 

43-95 

48 . . 

76.79 

42.22 

75 - 3 ° 

43- 1 3 

73-78 

43-98 

50 . . 

76.74 

42.25 

75-25 

43 - 16 

73-73 

44.01 

5 2 • • 

76.69 

42.28 

75.20 

43.18 

73.68 

44.04 

54 • • 

76.64 

42.31 

7 5- 1 5 

43.21 

73-63 

44.07 

56 . . 

. 76.59 

42-34 

75 - IQ 

43-24 

73-58 

44.09 

58 . . 

76.55 

42-37 

75-°5 

43-27 

73 - 5 2 

44.12 

60 . . 

76.50 

42.40 

75.00 

43-30 

73-47 

44.15 

1! 

p 

Ln 

0.66 

0.36 

0.65 

o -37 

0.65 

0.38 

c = 1.00 

0.88 

0.48 

0.87 

0-49 

0.86 

0.51 

c — 1.25 

1.10 

0.60 

1.09 

0.62 

1.08 

0.64 





















































STADIA TA CH YME TR Y. 


2 58 

This is a most generally useful stadia table for rods reading 
100 feet to the foot and with angles up to 30°. The values of 
other measures than those given in the table are obtained by 
multiplying the quantities under the proper vertical angle by 
stadia readings in hundreds of units. The quantity repre¬ 
senting the focal distance is very small and is given at the 
bottom of each page for focal lengths between f and ij feet 
and is represented as a constant equal to c, which corresponds 
with the second term in the right side of equations (6) and (7) 
(Art. 104). The direct use of the table involves a multiplica¬ 
tion for each result obtained. 

Example: Let rod intercept be 3.25 feet, and the angle 
of inclination be 5 0 35'. Then the distance on the horizontal 
would be 

d — 325 feet. 

If we accept the focal distance f c as 1.25 feet, we 
have from the tables and by substituting in formulas (10) 
and (11) 

d' — 3.25 ft. X 99 .°5 + 1.24 = 3 1 3 * 1 5 

and 

h =3.25 ft. X 9.68 + o. 11 = 31.57 ft. 

108. Determining Elevations by Stadia. —Table X, 
computed by Prof. R. S. Woodward, is one of the most 
convenient for determining differences of elevation from 
measures made with stadia. 

This table is computed from the formula 

h — d sin a cosin a\ .... (12) 

in which d is the observed distance of the rod, a is the angle 
of elevation or depression, and h is the difference of eleva¬ 
tion. To use the table, look for the observed angle in the 
first column, and the distance in the upper line under 


DIAGRAM FOR REDUCING STADIA MEASURES . 259 

the differences of elevation will be found at the intersection 
of the two columns. 

Example: Assuming the observed distance read directly 
from the stadia-rod as 360 feet and the angle 2° 40', we 
have from the table directly the result 

h — 16.7 feet. 

109. Diagram for Reducing Stadia Measures. —Stadia 
measures may be reduced by a diagram more simply, though 
not with the same degree of accuracy, as by tables. The 
following diagram (Fig. 78), from Ira O. Baker’s “ Engineer’s 



Surveying Instruments,” gives directly the corrections to 
horizontal distances in the upper horizontal line corresponding 
to the observed distances on the left-hand vertical line, and 
the angles indicated by diagonal intersecting lines. The 
right-hand vertical column of figures are differences of eleva¬ 
tion. These are found by the intersection of the lines of 








































2 6o 


STADIA TA CH YME TR Y. 


Table 

DIFFERENCES OF ELEVATION 


a 

/> 

100 

110 

I) 

120 

I) 

130 

J> 

140 

/> 

150 

I) 

160 

7 > 

170 

/> 

180 

I) 

190 

I> 

200 

/> 

220 

240 

I) 

260 

o / 

o OI 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

O. I 

O. I 

O. I 

O. I 

O. I 

O. I 

O 02 

O. I 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0 I 

O. I 

O. I 

O. I 

O. I 

O. I 

0.2 

o 03 

O. I 

0.1 

O. I 

0.1 

0.1 

0.1 

0.1 

0.1 

O 2 

0.2 

0.2 

0.2 

0.2 

0.2 

0 04 

O. I 

0.1 

O. I 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

o -3 

°-3 

0.3 

0 05 

O. I 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

°-3 

o -3 

°-3 

0.3 

0.3 

0.4 

0 06 

0.2 

0.2 

0.2 

0.2 

0.2 

o -3 

°-3 

o -3 

o -3 

o -3 

0.4 

0.4 

0.4 

o -5 

0 07 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

o -3 

o -3 

0.4 

0.4 

0.4 

0.4 

o -5 

o -5 

0 08 

0.2 

o -3 

o -3 

0.3 

0 3 

0.3 

0.4 

0.4 

0.4 

0.4 

o -5 

o -5 

0.6 

0.6 

0 og 

o -3 

o -3 

o -3 

o -3 

0.4 

0.4 

0.4 

0.4 

o -5 

o -5 

05 

0.6 

0.6 

0.7 

0 10 

°-3 

°-3 

°-3 

0.4 

0.4 

0.4 

0.5 

o -5 

o -5 

0.6 

0.6 

0.6 

0.7 

0.8 

0 11 

0.3 

0.4 

0.4 

0.4 

0.4 

o -5 

0 5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.8 

0.8 

0 12 

o -3 

0.4 

0.4 

o -5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0 13 

0.4 

°- 4 

o -5 

0.5 

0.5 

0.6 

0 6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

0 14 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

1.1 

0 15 

0.4 

0-5 

0.5 

0.6 

0.6 

0.7 

0.7 

°-7 

0.8 

0.8 

0.9 

1.0 

1.0 

1.1 

0 16 

o -5 

0-5 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.9 

°9 

1.0 

1.1 

1.2 

0 17 

o -5 

0-5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1.3 

0 18 

°-5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.2 

1 -3 

1.4 

0 19 

0.6 

o.6 

0.7 

0.7 

0.8 

0.8 

0.9 

o.g 

1.0 

1.1 

1.1 

1.2 

*•3 

1 -4 

0 20 

0.6 

0.6 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

x -3 

1.4 

i -5 

O 2 T 

0.6 

0.7 

0.7 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

i -3 

1 *5 

i .6 

O 22 

0.6 

0.7 

0.8 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

i -3 

i -4 

i -5 

1 • 7 

0 23 

0.7 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.1 

1.2 

»*3 

1 -3 

i -5 

1.6 

i -7 

0 24 

0.7 

0.8 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

*•3 

i *3 

1 • 4 

1 *5 

i -7 

1.8 

0 25 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1.2 

i -3 

1.4 

i -5 

i .6 

i -7 

1.9 

0 26 

0.8 

0.8 

0.9 

1.0 

1.1 

I. I 

1.2 

1 -3 

1.4 

i -4 

1-5 

1 • 7 

1.8 

2.0 

O 27 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1 • 3 

*•3 

1.4 

i -5 

1.6 

i -7 

1.9 

2.0 

0 28 

0.8 

0.9 

1.0 

1.1 

1.1 

1.2 

1 -3 

1.4 

i -5 

1 • 5 

1.6 

1.8 

2.0 

2.1 

O 29 

0.8 

0.9 

1.0 

1.1 

1.2 

i -3 

1.4 

1.4 

i -5 

1.6 

i -7 

1.9 

2.0 

2.2 

0 30 

0.9 

1.0 

1.0 

1.1 

1.2 

i -3 

1.4 

i -5 

1.6 

*•7 

i -7 

1.9 

2.1 

2.3 

0 35 

1.0 

1.1 

1.2 

i -3 

1.4 

i -5 

1.6 

i -7 

1.8 

1.9 

2.0 

2.2 

2.4 

2.6 

O 40 

1.2 

!-3 

1.4 

i -5 

1.6 

i -7 

1.9 

2.0 

2.1 

2.2 

2.3 

2.6 

2.8 

30 

0 45 

*•3 

1 -4 

1.6 

i -7 

1.8 

2.0 

2.1 

2.2 

2.4 

2.5 

2.6 

2.9 

3 -i 

3-4 

0 50 

1 5 

i .6 

i -7 

1.9 

2.0 

2.2 

2-3 

2.5 

2.6 

2.8 

2.9 

3-2 

3-5 

3-8 

0 55 

1.6 

1.8 

1.9 

2 . I 

2.2 

2.4 

2.6 

2.7 

2.9 

3 -o 

3-2 

3-5 

3-8 

4.2 

1 00 

1-7 

1.9 

201 

2.3 

2.4 

2.6 

2.8 

3 -o 

3 -i 

3-3 

3-5 

3-8 

4.2 

4-5 

I IO 

2 .O 

2.2 

2.4 

2.6 

2.9 

3 - 1 

3*3 

3-5 

3-7 

3-9 

4.1 

4-5 

4.9 

5-3 

I 20 

2-3 

2.6 

2.8 

30 

3-3 

3-5 

3-7 

4.0 

4.2 

4.4 

4-7 

5 -i 

5-6 

6.0 

1 30 

2.6 

2.9 

3 -i 

3-4 

3-7 

3-9 

4.2 

4.4 

4-7 

5 -o 

5-2 

5-8 

°-3 

6.8 

I 40 

2.9 

3-2 

3-5 

3-8 

4.1 

4.4 

4-7 

4-9 

5-2 

5-5 

5-8 

6.4 

7.0 

7.6 

1 50 

3-2 

3-5 

3.8 

4.2 

4-5 

4-8 

5 -i 

5-4 

5-8 

6.1 

6.4 

7.0 

7-7 

8-3 

2 00 

3-5 

3-8 

4.2 

4-5 

4-9 

5-2 

5-6 

5-9 

6-3 

6.6 

7.0 

7-7 

8.4 

9.1 

2 IO 

3-8 

4-2 

4-5 

4.9 

5-3 

5-7 

6.0 

6.4 

6.8 

7.2 

7.6 

8-3 

9.1 

9.8 

2 20 

4.1 

4-5 

4-9 

5-3 

5-7 

6 .1 

6.5 

6.9 

7-3 

7-7 

8.1 

8.9 

9.8 

10.6 

2 30 

4-4 

4.8 

5-2 

5-7 

6.1 

6-5 

7.0 

7-4 

7.8 

8-3 

8.7 

9.6 

10 .5 

Xl, 3 

2 40 

4.6 

5 • 1 

5-6 

6.0 

6.5 

7.0 

7-4 

7-9 

8.4 

8.8 

9-3 

10.2 

11.2 

12. I 

2 50 

4.9 

5-4 

5-9 

6.4 

6.9 

7-4 

7-9 

8.4 

8.9 

9.4 

9.9 

10.9 

11.8 

12.8 

3 00 

5-2 

5-7 

6-3 

6.8 

7-3 

7.8 

8.4 

8.9 

9.4 

9.9 

10.5 

”•5 

12.5 

13.6 

4 00 

7 *° 

7-7 

8.4 

9.0 

9-7 

10.4 

II . I 

11.8 

12.5 

13.2 

13-9 

* 5-3 

16.7 

l8. I 

6 00 

8.7 

9.6 

10.4 

11 -3 

12.2 

13.0 

* 3-9 

14.8 

15.6 

16.s 

T 7-4 

19.1 

20.8 

22.6 


I) 

J) 

D 

I) 

/> 

I) 

I) 

I) 

I) 

J) 

/> 

/> 

I) 

I) 


100 

110 

120 

130 

140 

150 

160 

170 

180 

190 

200 

220 

240 

260 


























































DIFFERENCES OF ELEVATION ,. 


2 bi 


X. 

FROM STADIA MEASURES. 


/> 

1 > 

I ) 

I ) 

I ) 

I ) 

1 ) 

I ) 

TJ 


1 ) 

I ) 

I ) 

J ) 

280 

300 

320 

340 

360 

380 

400 

420 

440 

460 

480 

500 

520 

640 

O. I 

0.1 

O. I 

O. I 

O. I 

O. 1 

O. I 

O. T 

O. I 

O. I 

0.1 

O. I 

0.2 

0.2 

O. 2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

o -3 

o -3 

0-3 

°-3 

°-3 

03 

O 2 

o -3 

o -3 

o -3 

0 3 

°-3 

o -3 

0 4 

0.4 

0.4 

0.4 

0 4 

o -5 

o -5 

°-3 

0.3 

0.4 

0.4 

0.4 

O.4 

0.5 

o -5 

o -5 

°-5 

0.6 

0.6 

0.6 

0.6 

0.4 

0.4 

0.5 

o-S 

o -5 

0.6 

0.6 

0.6 

0.6 

0.7 

°* 7 

0.7 

0.8 

0.8 

o -5 

°-5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

o.6 

0 6 

0.7 

0.7 

0.7 

0.8 

0.8 

°- 9 

0.9 

0.9 

I .O 

1.0 

1.1 

1 . 1 

c • 7 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1 0 

1.1 

I . I 

1.2 

1.2 

i -3 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

i -3 

1 .3 

1.4 

1.4 

o.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

i -3 

1 -3 

1.4 

i -5 

i -5 

i .6 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

i -3 

i -3 

1.4 

i-S 

i -5 

i .6 

1 7 

i -7 

I .o 

1 0 

1.1 

1.2 

i -3 

1-3 

14 

1.5 

i -5 

i. 6 

1 -7 

i -7 

1.8 

1.9 

I. I 

1.1 

1.2 

i -3 

1.4 

1.4 

i *5 

1.6 

i -7 

1 • 7 

1.8 

1.9 

2.0 

2.0 

I. I 

i .2 

i -3 

1.4 

1 • 5 

!-5 

i 6 

i -7 

1.8 

1.9 

2.0 

2.0 

2 . I 

2.2 

I .2 

1 *3 

1.4 

i -5 

i. 6 

!-7 

i -7 

1.8 

1.9 

2.0 

2.1 

2.2 

2-3 

2.4 

1 *3 

1-4 

i -5 

1.6 

i -7 

1.8 

1.9 

2.0 

2.0 

2.1 

2.2 

2-3 

2.4 

2-5 

1.4 

i -5 

1.6 

i -7 

1.8 

1.9 

2.0 

2 . I 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

1 • 5 

1.6 

1-7 

1.8 

1.9 

2.0 

2.1 

2.2 

2-3 

2.4 

2.5 

2.6 

2.7 

2.8 

*•5 

i -7 

1.8 

1.9 

2.0 

2.1 

2.2 

2-3 

2.4 

2-5 

2.7 

2.8 

2.9 

3 -o 

i.6 

i -7 

1.9 

2.0 

2.1 

2.2 

2-3 

2.4 

2.6 

2.7 

2.8 

2.9 

3.0 

3 -i 

1 • 7 

1.8 

2.0 

2.1 

2.2 

2.3 

2.4 

2.6 

2.7 

2.8 

2.9 

3 ■ 1 

3-2 

3-3 

i.8 

1.9 

2.0 

2.2 

2-3 

2.4 

2.6 

2.7 

2.8 

2.9 

3-1 

3-2 

3-3 

3-5 

1.9 

2.0 

2.1 

2.3 

2.4 

2-5 

2.7 

2.8 

2.9 

3 -i 

3-2 

3-3 

3-5 

3 - 6 

2,0 

2 . I 

2.2 

2.4 

2.5 

2.7 

2.8 

2.9 

3 - 1 

3-2 

3 4 

3-5 

3 - 6 

3-8 

2.0 

2.2 

2.3 

2-5 

2.6 

2.8 

2.9 

3-1 

3-2 

3-3 

3-5 

3 - 6 

3-8 

3*9 

2 . I 

2-3 

2.4 

2.6 

2.7 

2.9 

3 -o 

3-2 

3-3 

3-5 

3-6 

3-8 

3-9 

4.1 

2.2 

2.4 

2.5 

2.7 

2.8 

3 -o 

3 -i 

3-3 

3-5 

3 - 6 

3-8 

3-9 

4.1 

4.2 

2.3 

2.4 

2.6 

2.8 

2.9 

3 -i 

3-3 

3-4 

3-6 

3-7 

3-9 

4.1 

4.2 

4.4 

2.4 

2.5 

2.7 

2.9 

3 -o 

3-2 

3-4 

3-5 

3-7 

3-9 

4.1 

4.2 

4.4 

4.6 

2.4 

2.6 

2.8 

3 -o 

3 -i 

3-3 

3-5 

3-7 

CO 

CO 

4.0 

4.2 

4.4 

4-5 

4-7 

2.9 

3 -i 

3-3 

3-5 

3-7 

3-9 

4.1 

4-3 

4-5 

4 7 

4.9 

5 -i 

5-3 

5-5 

3-3 

3*5 

3-7 

4.0 

4.2 

4.4 

4-7 

4.9 

5 • 1 

5-3 

5-6 

5-8 

6.0 

6-3 

3-7 

3-9 

4.2 

4-5 

4-7 

5 -o 

5-2 

5-5 

5-8 

6.0 

6-3 

6-5 

6.8 

7-i 

4 • 1 

4.4 

4-7 

4-9 

5-2 

5-5 

5-8 

6.1 

6.4 

6.7 

7.0 

7-3 

7.6 

7-9 

4-5 

4.8 

5 -i 

5-4 

5-8 

6.1 

6.4 

6.7 

7.0 

7-4 

7-7 

8.0 

8-3 

8.6 

4.9 

5-2 

5-6 

5-9 

6 -3 

6.6 

7.0 

7-3 

7-7 

8.0 

8.4 

CO 

9.1 

9.4 

5-7 

6.1 

6-5 

6.9 

7-3 

7*7 

8.1 

8.6 

9.0 

9.4 

9.8 

10.2 

10.6 

11.0 

6-5 

7.0 

7-4 

7-9 

8.4 

8.8 

9-3 

9.8 

10.2 

10.7 

I I .2 

11.6 

12.1 

12.6 

7*3 

7-9 

8.4 

8.9 

9.4 

9.9 

10 .5 

I I .O 

11 -5 

12.0 

12.6 

’ 3 - 1 

13.6 

14. I 

8.1 

8.7 

9-3 

9.9 

10.5 

II .O 

11.6 

12.2 

12.8 

' 3-4 

14.0 

M .5 

15.1 

15-7 

9.0 

9.6 

10.2 

10.9 

11.5 

12.2 

12.8 

J 3-4 

14.1 

14.7 

15-4 

16.0 

16.6 

* 7-3 

vO 

CO 

10.5 

II .2 

11.9 

12.6 

* 3-3 

14.0 

14.6 

T 5*3 

16.0 

16.7 

17.4 

l8. I 

18.8 

10.6 

11.3 

12 . I 

12.8 

13.6 

14.4 

15-1 

15-9 

16.6 

17.4 

18.1 

18.9 

19.6 

20.4 

11.4 

12.2 

13.° 

13.8 

14.6 

i.S -5 

16.3 

17. j 

17.9 

18.7 

19-5 

20.3 

21.2 

22.0 

12.2 

13- 1 

13-9 

14 8 

*5 • 7 

16.6 

17.4 

18.3 

T9.2 

20.0 

20.9 

21 8 

22.7 

23 5 

13.° 

J 3‘9 

14.8 

15-8 

16.7 

17.7 

18.6 

> 9-5 

20.5 

21.4 

22.3 

23.2 

24.2 

25-1 

13.8 

14.8 

15-8 

16.8 

17.8 

18.8 

19.7 

20.7 

21.7 

22.7 

23-7 

24.7 

2 5*7 

26.7 

14.6 

15-7 

16.7 

^4 

00 

18.8 

x 9-9 

20.9 

21.9 

23.0 

24.O 

25-1 

26.1 

27.2 

28.2 

19.5 

20.9 

22.3 

23 -7 

25.1 

26.4 

27.8 

29.2 

20 6 

32.0 

33-4 

34-8 

36.2 

37-6 

24-3 

26.0 

27.8 

29-5 

3 i -3 

33 -o 

34-7 

36 -5 

38.2 

39 9 

41.7 

43-4 

45 •> 

46.9 

1 ) 

1 ) 

I ) 

7 > 

I) 

7) 

1 ) 

J) 

D 

J) 

I ) 

I) 

1 > 

2 > 

280 

300 

320 

340 

360 

380 

400 

420 

440 

460 

480 

O 

O 

*•5 

520 

640 



















































262 


A TA DI A TA CH YME TR Y. 


Table 

DIFFERENCES OF ELEVATION 


a 

7 > 

56 < 

D 

58 C 

JJ 

60 C 

/> 

62 C 

I) 

640 

/> 

660 

JJ 

680 

n 

700 

1 J 

720 

JJ 

740 

JJ 
j 760 

JJ 

780 

JJ 

800 

JJ 

820 

o 

o 

/ 

OI 

0.5 

0.5 

0.2 

0.5 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

O 2 

o 

02 

0.5 

o-2 

°-3 

0.* 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

°-5 

o 

03 

O.C 

o-' 

0.5 

0 •; 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

°- 7 

0.7 

0.7 

0.7 

o 

04 

0.6 

0.7 

o. 7 

o -7 

O.7 

0.8 

D. 8 

0.8 

0.8, 

0.9 

0.9 

°-9 

0.9 

I .O 

0 

05 

0.8 

o.$ 

0.9 

o.c 

0.9 

1.0 

1.0 

1.0 

1.0 

1 . 1 

I . I 

I . I 

1.2 

| 1.2 

o 

06 

I .0 

I .0 

1 . 1 

I . I 

I . I 

1.2 

1.2 

1.2 


1 -3 

*•3 

1.4 

1.4 

7-4 

o 

07 

1 . 1 

1.2 

1.2 

1 *3 

*•3 

1 *3 

1.4 

1.4 

i -5 

*•5 

1.6 

i. 6 

1.6 

7-7 

o 

08 

1-3 

1.4 

1.4 

1.4 

i -5 

1 • 5 

i .6 

1.6 

1 • 7 

* 7 • 7 

1.8 

1.8 

7.9 

1.9 

o 

09 

1 • 5 

1 -5 

1.6 

1.6 

i -7 

i -7 

1.8 

1.8 

1.9 

1.9 

2.0 

2.0 

2.1 

2.1 

0 

10 

1.6 

7-7 

1 • 7 

1.8 

i -9 

1.9 

2.0 

2.0 

2.1 

2 . 2 

2.2 

2-3 

2-3 

2.4 

o 

II 

1.8 

1 .0 

1 *9 

2.0 

2.0 

2.1 

2.2 

2.2 

2-3 

2.4 

2.4 

2-5 

2.6 

2.6 

o 

12 

2.0 

2.0 

2. I 

2.2 

2.2 

2-3 

2.4 

2.4 

2.5 

2.6 

2.7 

2.7 

2.8 

2.9 

o 

*3 

2. I 

2 . 2 

2-3 

2-3 

2.4 

2-5 

2.6 

2.6 

2.7 

2.8 

2.9 

2.9 

3 -o 

3-7 

o 

14 

2.3 

2-4 

2.4 

2.5 

2.6 

2.7 

2.8 

2.8 

2.9 

3-0 

3 -i 

3-2 

3-3 

3-3 

0 

15 

2.4 

2-5 

2 . 6 

2.7 

2.8 

2.9 

3 -o 

3 * 1 

3 * 1 

3"2 

3-3 

3*4 

3-5 

3.6 

o 

l6 

2.6 

2.7 

2.8 

2.9 

3 -o 

3 - 1 

3-2 

3-3 

3-3 

3-4 

3-5 

3-6 

3-7 

3-8 

o 

17 

2.8 

2.9 

3 -o 

3 -i 

3-2 

3-3 

3-4 

3-5 

3-6 

3-7 

3.8 

3-9 

4.0 

4.1 

o 

18 

2 .9 

3 ,Q 

3 - 1 

3-2 

3-4 

3-5 

3-6 

3-7 

3-8 

3-9 

4.0 

4-7 

4.2 

4-3 

o 

*9 

3 - 1 

3-2 

3-3 

3-4 

3-5 

3-6 

3-8 

3-9 

4.0 

4 -i 

4.2 

4-3 

4.4 

4-5 

0 

20 

3-3 

3-4 

3-5 

3-6 

3-7 

3-8 

4.0 

4.1 

4.2 

4-3 

4.4 

4-5 

4-7 

4.8 

o 

21 

3-4 

3-5 

3-7 

3-8 

3-9 

4.0 

4.2 

4-3 

4.4 

4-5 

4.6 

4.8 

4-9 

5 -o 

o 

22 

3-6 

3-7 

3 - 8 

4.0 

4 -i 

4.2 

4.4 

4-5 

4.6 

4-7 

4-9 

5.0 

S-i 

5-2 

o 

23 

3-7 

3-9 

4.0 

4.1 

4-3 

4.4 

4-5 

4-7 

4.8 

5 -o 

5 -i 

5-2 

5-4 

5-5 

o 

24 

3-9 

4.0 

4.2 

4-3 

4-5 

4.6 

4-7 

4.9 

5 -o 

5-2 

5-3 

5-4 

5.6 

5-7 

0 

25 

4.1 

4.2 

4.4 

4-5 

4-7 

4.8 

4.9 

5 -i 

5-2 

5-4 

5-5 

5-7 

5.8 

6.0 

o 

26 

4.2 

4.4 

4-5 

4-7 

4.8 

5 -o 

5 -i 

5-3 

5-4 

5-6 

5-7 

5-9 

6.0 

6.2 

o 

27 

4-4 

4.6 

4-7 

4.9 

5 -o 

5-2 

5-3 

5-5 

5-7 

5-8 

6.0 

6.1 

6-3 

6.4 

o 

28 

4.6 

4-7 

4.9 

5 -o 

5-2 

5-4 

5-5 

5-7 

5-9 

6.0 

6.2 

6-3 

6-5 

6.7 

o 

29 

4 7 

4.9 

5 -i 

5-2 

5-4 

5.6 

5-7 

5-9 

6.1 

6.2 

6.4 

6.6 

6.8 

6.9 

0 

30 

4.9 

5 -i 

5-2 

5-4 

5.6 

5-8 

5-9 

6.1 

6.3 

6 *5 

6.6 

6.8 

7.0 

7-2 

o 

35 

5-7 

5-9 

6 . 1 

6-3 

6 -5 

6.7 

6.9 

7-1 

7-3 

7-5 

7-7 

7-9 

8.1 

8.4 

o 

40 

6-5 

6.7 

7.0 

7.2 

7-4 

7-7 

7-9 

8.1 

8.4 

8.6 

8.8 

9.1 

9-3 

9-5 

o 

45 

7-3 

7.6 

7-9 

8.1 

6.4 

8.6 

8.9 

9.2 

9.4 

9-7 

9.9 

10.2 

10.5 

IO.7 

o 

50 

8 . 1 

8.4 

8.7 

9.0 

9-3 

9.6 

9 9 

10.2 

10.5 

10.8 

I I . I 

77.3 

11.6 

11.9 

o 

55 

9.0 

9-3 

9.6 

9.9 

10.2 

10.6 

IO.9 1 

I I .2 

n -5 

11.8 

12.2 

72.5 

12.8 

73.7 

1 

00 

9.8 

IO. I 

10.5 

10.8 

II .2 

n -5 

11. 9 J 

12.2 

12.6 

12.9 

* 3-3 

7 . 3-6 

14.0 

74-3 

I 

IO 

11.4 

11.8 

12.2 

12.6 

13-0 

13-4 

13-8 

*4.3 

14.7 

i 5 -i 

75-5 

75-9 

16.3 

16.7 

I 

20 

13.0 

* 3*5 

14.0 

H -4 

14.9 

* 5-4 

15-8 

16.3 

16.7 

17.2 

17.7 

18.1 

18.6 

19.1 

I 

30 

r 4 • 7 

15.2 

l 5 • 7 

16.2 

16.7 

17-3 

17.8 

18.3 

18.8 

19.4 

>9 9 

2° .4 

20 . Q 

21 5 

I 

40 

16.3 

16.9 

17.4 

18.0 

18.6 

19.2 

19.8 

20.3 

20.9 

ai *5 

22 . I 

22 . 7 

23-3 

23.S 

I 

50 

17.9 

18.5 

19.2 

19.8 

20.5 

21 . I 

21.7 

22.4 

23.0 

2 3-7 

24-3 

24.9 

25.6 

26.2 

2 

00 

19-5 

20.2 

20.9 

21.6 

22.3 

23.0 

23-7 

24.4 

25.1 

25.8 

26.5 

27.2 

27.9 

28.6 

2 

10 

21.2 

21 .9 

22 . 7 

23-4 

24.2 

24.9 

25-7 

26.4 

27.2 

28.0 

28.7 

29.5 

30.2 

31.0 

2 

20 

22.8 

23.6 

24.4 

25.2 

26.0 

26.8 

27.7 

28.5 

29-3 

3 °. 1 

3°-9 

31.7 

32.5 

33-4 

2 

30 

24.4 

25-3 

26.1 

27.0 

27.9 

28.8 

29.6 

3 o -5 

31-4 

32.2 

33 -i 

34-0 

34-9 

35-7 

2 

40 

26.0 

27.O 

27.9 

28.8 

29.7 

307 

31.6 

32.5 

33-5 

34-4 

35-3 

36.3 

37-2 

38.1 

2 

50 

27.6 

28.6 

29.6 

30.6 

31.6 

32.6 

33-6 

34-6 

35-5 

36.5 

37-5 

38.5 

39-5 

40.5 

3 

00 

29-3 

30-3 

3>-4 

32.4 

33-4 

34-5 

35-5 

36.6 

37-6 

38.7 

39-7 

40.8 

41.8 

42.9 

4 

00 

39.0 

40.4 

41.8 

43 - 1 

44-5 

45-9 

47-3 

48.7 

50.1 

5 i -5 

52.9 

54-3 

55-7 

57 -i 

5 

00 

*8.6 

50.4 

52.1 

53-8 

55-6 | 

57-3 

59 -o 

60.8 

62.5 

64.2 

66.0 

67.7 

69-5 

71.2 



1) 

]) 

JJ 

IJ 

/> 

JJ 

I) 

/> 

IJ 

JJ 

JJ 

1 > 

/> 

n 



560 

580 

600 

6*20 

610 

660 

680 

700 

720 

740 

760 

780 

800 

820 





































































































DIFFERENCES OF ELEVATION. 


263 


X. 

FROM STADIA MEASURES. 


I ) 

840 

J ) 1 > 
800 880 

D 

000 

I 

Z> 

920 

J ) i 
940 

1 ) 

960 

I > 

980 

l ) 

1000 

1 ) 

1100 

I ) 

1200 

1 

T ) 

1 :ioo 

J ) 

1400 

J> i 

1500 

1 > 

2000 

0.2 

0.2 

o -3 

o -3 

o -3 

o -3 

o -3 

o -3 

o -3 

o -3 

0 3 

0.4 

04 

0.4 

0 6 r 

o *5 

o -5 

0-5 

o -5 

o -5 

0.5 

0.6 

0.6 

0.6 

0.6 

0.7 

0.8 

0.8 

0.9 

7 2 | 

0.7 

0.7 

0.8 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

1.0 

1 O 

1.1 

1.2 

1 -3 

1 -7 1 

I .O 

1.0 

I .O 

1.0 

1.1 

I . I 

1.1 

1.1 

1.2 

x -3 

1.4 

15 

1.6 

' -7 

2 -3 | 

I .2 

I .2 

7-3 

1 -3 

1 -3 

1.4 

1.4 

x -4 

x -5 

i. 6 

J -7 

x -9 

2.0 

2.2 

2.9 

*•5 

i -5 

1 • 5 

i .6 

i .6 

1.6 

J -7 

x -7 

x -7 

1.9 

2. I 

2-3 

2.4 

2.6 

3-5 

J -7 

1.8 

1.8 

1.8 

i -9 

x -9 

2.0 

2 . O 

2.0 

2.2 

2.4 

2.7 

2.91 

31 

4 -J 

2.0 

2.0 

2.1 

2.1 

2. I 

2.2 

2.2 

2-3 

2-3 

2.6 

2.8 

3 -o 

3-3 

3-5 

4-7 

2.2 

2-3 

2-3 

2-4 

2.4 

2.5 

2.5 

2 6 

2.6 

2.9 

3 - 1 

3-4 

3-7 

3-9 

5-2 

2.4 

2-5 

2.6 

2.6 

2.7 

2.7 

2.8 

2.9 

2.9 

3-2 

3-5 

3-8 

4 -J 

4-4 

5.8 

2.7 

2.8 

2.8 

2.9 

2.9 

30 

3 • 1 

3 -i 

3.2 

3-5 

3-8 

4 2 

4-5 

4-8 

6.4 

2.9 

3.0 

3 • 1 

3 -i 

3-2 

3-3 

5-4 

3-4 

3-5 

3-8 

4.2 

4-5 

4-9 

5-2 

7.0 

3 - 2 

3-3 

3-3 

3-4 

3-5 

3-6 

3 - 6 

3-7 

3-8 

4 2 

4-5 

4.9 

5-3 

5-7 

7.6 

3-4 

3-5 

3 - e 

3-7 

3-7 

3-8 

3-9 

4.0 

4 -x 

4-5 

4-9 

5-3 

5 7 

6.1 

8.1 

3-7 

3-7 

3-8 

3-9 

4.0 

4 .x 

4.2 

4-3 

4.4 

4-8 

5-2 

5-7 

6.1 

6.5 

8-7 

3-9 

4.0 

4 • 1 

4.2 

4-3 

4-4 

4-5 

4.6 

4-7 

5 - 1 

5-6 

6.0 

6-5 

7.0 

9-3 

4.2 

4-3 

4 4 

4-5 

4.6 

4-7 

4.8 

4 9 

5 -° 

5-4 

5-9 

64 

6.9 

7-4 

9-9 

4-4 

4-5 

4.6 

4-7 

4.8 

4.9 

5 -o 

5 • 1 

5-2 

5-8 

fl • 3 

6.8 

7-3 

7-9 

10.5 

4.6 

4.8 

4-9 

5 -° 

5 - 1 

5-2 

5-3 

5-4 

5 * 5 

6.1 

6.6 

7.2 

7-7 

8-3 

I I . I 

4-9 

5 -o 

5 -i 

S- 2 

5-4 

5-5 

5-6 

5-7 

5-8 

6.4 

7.0 

7-5 

8.1 

8-7 

11.6 

5 • 1 

5-3 

5-4 

5-5 

5-6 

5-7 

5-9 

6.0 

6.1 

6.7 

7-3 

7-9 

8.6 

9.2 

12.2 

5-4 

5-5 

5-6 

5-8 

5-9 

6.0 

6.1 

6-3 

6.4 

7 -° 

7-7 

8-3 

9.0 

9.6 

12.8 

5-6 

5-8 

5-9 

6.0 

6.2 

6-3 

6.4 

6.6 

6.7 

7-4 

8 0 

8-7 

9 4 

10.0 

J 3-4 

5-9 

6.0 

6.1 

6.3 

6.4 

6.6 

6.7 

6.8 

7.0 

7-7 

8. 4 

9 -J 

9.8 

JO.5 

14.0 

6.1 

6.3 

6.4 

6.5 

6.7 

6.8 

7 -° 

7 -i 

7-3 

8.0 

8.7 

9-5 

10.2 

10.9 

J 4-5 

6.4 

6.5 

6.7 

6.8 

7.0 

7-1 

7-3 

7-4 

7.6 

8.3 

9 -J 

9.8 

10.5 

JJ -3 

J 5 -J 

6.6 

6.8 

6.9 

7 • 1 

7-2 

7-4 

7-5 

7-7 

7-9 

8.6 

9 4 

10.2 

1 1 .O 

j i.8 

J 5-7 

6.8 

7.0 

7.2 

7-3 

7-5 

7 7 

7-8 

8.0 

8.1 

9.0 

9-7 

10.6 

11 4 

12.2 

16.3 

7.1 

7-3 

7-4 

7.6 

7.8 

7-9 

8.1 

. 8.3 

8.4 

9-3 

IO. I 

I I .0 

ii .8 

J 2 - 7 - 

16.9 

7-3 

7-5 

7-7 

7-9 

8.0 

8.2 

8.4 

8.6 

8.7 

9.6 

10.5 

JJ -3 

12.2 

J 3 -J 

17 5 

8.6 

8.8 

9 -° 

9.2 

9.4 

9.6 

9 8 

IO.O 

10.2 

I I .2 

12.2 

13.2 

'4 3 

J 5-3 

20.4 

9.8 

10.0 

| 10.2 

10.5 

10.6 

10.9 

11.2 

11.4 

11.6 

12.8 

I4.O 

15 -J 

16.3 

J 7-4 

23 3 

I I .O 

1 1 • 3 

IX .5 

1 I*- 8 

12.0 

12.3 

12.6 

12.8 

13-1 

14 4 

1 5 ■ 7 

17.0 

18.3 

19.6 

26.2 1 

12.2 

■2-5 

12.8 

* 3 - 1 

13-4 

13-7 

14 . O 

14.2 

14-5 

16.0 

17.4 

18. q 

20.3 

21.8 

29. I 

13.4 

13.8 

i 4 -i 

14.4 

J 4 • 7 

15.0 

* 5-4 

15-7 

16.0 

17.6 

T9.2 

20.8 

22.4 

24.0 

32.0 

14.7 

150 

I5 ' 4 

15-7 

16.1 

16.4 

16.8 

I 7 - 1 

17-5 

I9.2 

20.9 

22.7 

24.4 

1 26.2 

34-9 

17.1 

17-5 

17.9 

18.3 

18.7 

19.1 

* 9-5 

20 O 

20.4 

22.4 

24.4 

26.5 

28.5 

i °-5 

40.7 

■ 9-5 

20.0 

20.5 

20.9 

21.4 

21.9 

22.3 

22.8 

23-3 

25.6 

27.9 

30.2 

32.6 

34-9 

46.5 t 

22.0 

22. S 2T.O 

23.6 

24.x 

24.6 

25-1 

25.6 

26.2 

28.8 

3 1 -4 

34 -o 

36.6 

39-3 

52-3 [ 

24.4 

25 O 

23.6 

26.2 

26.7 

27-3 

27.9 

28.5 

29.1 

32.0 

3 4 9 

37-8 

4 °.7 

43- 6 

58.1 

26.9 

27.5 28.1 

28.8 

29.4 

30.1 

30-7 

3 r -3 

32.0 

35-2 

38-4 

41.6 

44.8 

48.0 

64.0 

29-3 

3 °.° 

30.7 

3 1 *4 

32.1 

32.8 

33-5 

34-2 

34-9 

38-4 

4 T -9 

45-3 

48.8 

52 • 3 

69.8 

31 ■ 7 

32 -5 

33-2 

34 -o 

34-8 

35-5 

36-3 

37 -o 

37-8 

4x. 6 

45-3 

49.x 

52.9 

56.7 

75-6 

34.2 

35 -o 

35-8 

36.6 

37-4 

38.2 

39 - 1 

39-9 

40.7 

44-7 

48.8 

5 2 -9 

57 -o 

61.0 

81.4 

36.6 

37-5 

[38 4 

39-2 

40. I 

41 .O 

41.8 

42.7 

43 - 6 

47 9 

52.3 

56.7 

61 0 

65.4 

87.2 

39 -° 

40.0 

40.9 

41.8 

42.8 

43-7 

44.6 

45.6 

46.5 

51 • I 

55.8 

60.4 

65.1 

67.7 

93 -° 

4 1 -5 

42 . 5 , 43-4 

44-4 

45-4 

46.4 

47-4 

48.4 

49-4 

54-3 

59-2 

64.2 

69.1 

74 -i 

98.7 

43-9 

44.9 

46.0 

47.0 

48.1 

49.1 

50.2 

512 

52.3 

57-5 

62.7 

67.9 

73.2 

78.4 

104.5 

58 5 

=;q.8 61.2 

62.6 

64.0 

65-4 

66.8 

68.2 

6q.6 

76.5 

83-5 

90.5 

97-4 

104.4 

139.2 

72 9 

74-7 

76.4 

78.1 

79-9 

81.6 

83-3 

85.1 

86.8 

95-5 

104 2 

112.9 

121.5 

13 0 .2 

173.6 

1) 

/> 

J> 

/> 

]) 

J> 

1) 

J) 

J) 


/> 

J > 

T) 

I ' 

7 > 

J > 

840 

800'880 

1 1 

900 

920 

940 

1 

j 900 

1 

980 

1000 

1100 

1 

1200 

1300 

1 

1400 

1 

1500 

2000 





































































































264 


.S’ TA DIA TA CH YME TR Y. 


horizontal distances indicated in the lower margin, with the 
diagonal lines corresponding to angles of elevation. 

Example: Let rod intercept be 3.60 feet, and the angle 
2° 4c/; then it will be seen that the correction to the hori¬ 
zontal distance is too small to note on the diagram. The 
difference of elevation comes from the intersection of a 
vertical line between 300 and 400, and the right diagonal 
between 2° and 3 0 , and is approximately 16 to 18 feet. 

no. Diagram for Reducing Inclined Stadia Distances 
to Horizontal. —The following diagram (Fig. 79), prepared 
by Prof. Ira O. Baker, gives with some accuracy the correc¬ 
tion to the horizontal distance corresponding with any 
observed angle of inclination in the stadia-rod. The observed 
distance is indicated on the lower and right-hand marginal 
lines, the angle of inclination by the intersecting diagonal 
lines, and the correction to the horizontal distance, always 
minus, is given on the upper and left-hand marginal lines. 

Another though more complicated diagram for the same 
purpose is that illustrated in Fig. 80, designed by Mr. E. 
McCulloch and published in Engineering News. This diagram 
has a wide range and is well suited to the most detailed 
work. On the lower and left-hand outer margins are figured 
stadia readings in feet or meters, decimals of the same being 
interpolated on the inner margin on all four sides, where the 
angles of inclination are also indicated. Corrections to 
observed distances are found at the intersections of the ver¬ 
tical rod readings with the horizontal angle lines, or vice 
versa , the horizontal rod lines with the vertical angle lines, 
and by following out to the margins the diagonals at which 
these intersections occur; opposite the ends of the diagonals 
will be found the corrections in feet to the distances observed, 
such corrections being on all four outer margins. 

This correction is always to be subtracted from the dis¬ 
tance and is applied in the following manner: If the angle 
appears on the left margin, multiply the correction by 0.01 ; 


REDUCING INCLINED STADIA DISTANCES. 



Fig. 79 . —Stadia Reduction Diagram to Horizontal Distances. 






























































































266 


S TAD I A TA CIIYME TR Y. 


if on the top margin, multiply the correction by o.i ; if on 
the bottom margin, multiply the correction by i.O; and if on 
the right-hand margin, multiply the correction by io. 


100 

90 

80 

70 

00 

50 

10 


30 


20 


10 

Fig. 8o.—Diagram for Reducing Inclined Stadia Distances to Hori¬ 
zontal. 


50 60 70 80 90 100 



35- 


15° 16° 17° 18° 


9 10 


hi. Effects of Refraction on Stadia Measurements.— 

Experiments by Mr. J. L. Van Ornum in his stadia-work 
showed that thd disturbing effect of refraction increased 
enormously toward the ground, even if the foot of the rod 
were unobstructed. When no error in observation is made 
and the refraction remains constant from the time of the 























































































































































































































































EFFECTS OF REFRACTION ON STADIA. 267 

foresight to the time of the backsight, the elevation of any 
point can be computed by the formula 

H' — H -f- i(/i -f- m + n — //')., . . . (13) 

in which H — elevation of known station; ’ 

h — height of instrument at that station; 
in — total vertical component of the foresight; 

H r — elevation of the unknown station; 
ti — height of the instrument at the unknown 
station; and 

n — the total vertical component of the backsight. 

The average error of closing in Mr. Van Ornum’s work 
before the adoption of this formula was more than half 
greater than after its use. 

The effect of differential refraction on the determination 
of stadia distances is well set forth in a paper on stadia 
measurements by Mr. Leonard S. Smith of the University 
of Wisconsin. In these experiments Mr. Smith ascertained 
that refraction is a variable quantity, dependent on variable 
temperature of air and ground, and that it is much greater 
near the ground than 3 feet above it; also greater at noon 
than before or after it; that the effects vary for different dis¬ 
tances and also for different observers. He called differential 
refraction that due to the different amount of refraction on 
the line of sight of the upper stadia-hair and that on the 
lower stadia-hair. He further found that the effects of 
refraction accumulated as distance increased. Twelve miles 
of stadia measurements with centers averaging 600 feet, in 
the morning and evening hours, showed an accuracy of plus 
1 in 2685. The same distances measured by centers at 
midday showed an accuracy of minus 1 in 655. Again, 30 
miles of measurements made in the morning and evening 
hours with centers between one and two thousand feet 


268 


STADIA TA CH YME TR Y. 


showed an accumulated error of but I in 1741, while the 
same distance, in midday, developed an accumulative error 
of 1 in 289. 

The practical results of these experiments may be taken 
chiefly as suggestions, and the most interesting deductions to 
be obtained therefrom are: 

1. To obtain accuracy in stadia-work it is best to obtain 
an interval error of the rod for the effect of refraction at 
different hours of the day; and 

2. This correction for refraction may be made to readings 
for the hours corresponding to the refraction ascertained, or 
the stadia-rod may be graduated in proportion to the dis¬ 
tances of the various intercepts above the ground; this latter 
method is not recommended for its simplicity, however. 

It is evident that at midday long readings which require 
the lower stadia-wire to be lower than 3 feet from the bottom 
of the rod should not be taken. Where a rod interval is 
determined for correction of refraction, that interval which is 
determined for summer months or midday should not be used 
in colder months or winter, without testing by an independ¬ 
ent interval. Moreover, such interval should not be deter¬ 
mined for ordinary soil when the work is to be conducted 
over curbstones, in cities. 

To sum Mr. Smith’s results it may be stated that the 
time of greatest atmospheric vibration is about the middle of 
the forenoon, when the maximum difference in temperature 
occurs between atmosphere and ground; the stadia interval 
should be determined, not at any particular time of day, but 
at many hours during the day; and such interval for the 
hours should be selected as will approximate as closely as 
possible the average field conditions. Perhaps one of the 
simplest ways of applying the interval is not by dividing the 
rod unequally by incorporating the interval in the rod 
divisions, but by using a rod divided to standard lengths and 
computing distances by means of an interval factor. 


STADIA-RODS. 



112 . Stadia-rods. —Several methods of dividing stadia- 
rods for very accurate work have been used, such as making 
special subdivisions of the rod to correspond with distances 
subtended between fixed hairs, or of dividing the rod irregu¬ 
larly so as to incorporate within the divisions a stadia interval 
which will correct the effects of refraction. The most 
approved practice is, however, to use rods of standard divi¬ 
sion and to prepare tables, or, better still, a rod interval 
factor to be applied to the observed distance as a correction 
The following are some of the disadvantages of using other 
than standard rods: 

1. Subsequent tests for interval cannot be made without 
the expense of repainting and regraduating the rods; 

2. Rods specially divided cannot be interchanged among 
instruments; such rods cannot be used in leveling without 
computing the necessary correction; 

3. Leveling-rods cannot be used in stadia-work; and 

4. Observers with different personal equations cannot use 
the same rods without causing systematic errors in the work. 

Stadia-rods are of two general forms: 

1. Target; and 

2. Self-reading. 


The former is the only one which can be satisfactorily 
used in taking very long sights; the latter is most satisfactorily 
employed on short sights. Target-rods are similar to the 
simpler forms of leveling-rods, as the Philadelphia rod (Fig. 
96). Self-reading or speaking rods may be of the type of the 
Philadelphia rod, for short distances, but are more satisfac¬ 
torily made of flat boards 10 to 15 feet in length, 4 inches 
wide, and f to $ inch thick, of well-seasoned pine. These 
can be graduated by the topographer or by some painter after 
one of the numerous patterns which have been found satis¬ 
factory under various circumstances. Such forms of gradua¬ 
tion are better than the ordinary self-reading leveling-rods, 
because as a rule divisions on the latter are small and the 


2yo 


S TA DI A TA CH YME TR Y. 


figures small and they are therefore difficult to read at long 
distances. As visibility is the first requisite in a good stadia- 
rod, the graduations should consist of a number of divisions 
so large and yet of such varying shapes as to make them 
readily distinguishable at long distances, the pattern being 
either painted or stenciled on the wood or else on canvas or 
paper which may be fastened to the rod with glue, varnish, 
or both. 

In Fig. 81 are shown two of the best forms of rod 



a b 

’Fig. 8i.—Speaking Stadia and Level-rods. 

graduation for long sights. Rod (a) is well adapted to either 
metric or foot graduation, but in its use care must be exer¬ 
cised to accurately count the units, as these are not numbered. 
Rod ( 3 ) has the feet only numbered, but the figures are so 
large as to be readily legible at very considerable distances. 
The forms of the figures on rod (o) are such as to clearly 
define the tenths. 

















STADIA-RODS . 


27I 


In Fig. 97 are shown patterns of speaking level-rods, and 
these are also well adapted to stadia-work at short distances. 
All are suited to either metric or foot graduation. Rod (a) 
is perhaps best suited to close, detailed work. Rod {b) has 
the clearest figures and is therefore best suited to longer dis¬ 
tances. Rod ( c ) has the advantage of having odd and even 
tenths figured on opposite sides of the center of the rod, with 
the order changed for units so as to bring their numberings 
nearer the rod center; moreover, the graduations are less 
liable to injury because they are in the center of the rod. 
For rough work at long distances a pole may be cut and 
white string or cloth tied around it at every foot to serve as 
graduation. 


CHAPTER XIII. 


ANGULAR TACHYMETRY. 

113. Angular Tachymetry with Transit or Theodolite. 

—The angular system of tachymetry has the advantages— 

1. Of enabling the survey to be made by the ordinary 
transit without the addition of stadia-wires; 

2. That there are no divisions of the rod to be read; and 

3. That, because of the ease and simplicity of the obser¬ 
vations, very long sights can be taken with a small telescope 
with great accuracy. 

The field-work by this method is nearly as rapid as that 
by any other form of tachymetry, as there are but two angles 
to be read, as against one angle and a rod reading with stadia 
method. The reduction of the work is less simple and more 
laborious than that by means of stadia measurements. That 
this method is as accurate as the stadia method appears from 
experiments made by Messrs. Airy and Middleton in England. 
The former used a five-inch theodolite and an ordinary level¬ 
ing-rod 16 feet in length. He ran a circuit 1^- miles in 
length, involving differences in elevation of 118 feet: the 
average length of stations was 341 feet, and the resulting 
probable error of distance was 3.64 feet in a mile, and of 
leveling 0.033 feet. The average closure error of four series 
of measurements was 2.84 feet. According to Mr. Middleton 
the limit of accuracy in this method is reached when the rod 
is held at a distance of 1000 feet, as determined from his 


272 


ANGULAR TACHYMETRY WITH TRANSIT. 


273 


measurements, and the average error in one mile was found to 
be 6.15 feet. 

The accompanying diagram (Fig. 82) illustrates the method 



of angular tachymetry with a transit or theodolite. For 
simplicity the rod should be unmarked excepting for two or 
three broad lines painted at known distances apart, one near 
the bottom, one near the top, and one near the middle. These 
lines, if properly proportioned, can be seen at distances as 
great as a mile, and thus permit of tachymetric measurements 
of moderate accuracy for that distance. 

Let S and S' = distance on the rod between two well- 

defined marks? 

h and ti = height of the lower marks above the 

ground ; 

H and //' = height of the lower marks above line of 

collimation of instrument; 

e and e — height of ground surface at foot of rod 

above base or ground surface at instru¬ 
ment ; 

i — height of collimation of instrument above 
ground surface; 

a and a' = angle at instrument between horizon and 

lower mark; 

/3 and /?' = angle at instrument between horizon and 

upper mark; 

d and d' = horizontal distance from instrument to 

rods;—then 


5 = d(tan — tan a) .(14) 









2 74 


ANGULAR TACHYMETRY. 


Transposing, we have 

d = 


S 


also 


tan ft — tan ot ’ 
II —d tan 


• • 


Substituting, we have 


H = 


S tan ac 


tan ft — tan ol 


• • (is) 

• • 0 6 ) 

• • (17) 


It is also seen from an inspection of the figure that 
approximately 

€ — H — ll —j— z. ..... (l8) 

From these the height of the ground at observed stations 
can be reduced to sea-level by addition of the height of 
ground at instrument. The above computations are easily 
made with the aid of a table of natural functions (Tables XI 
and XII). //and//' must have their proper signs, and will 
result in giving the difference of level according as the new 
stations are above or below the position of the instrument. 
From the expression dH it may be shown that a small angu¬ 
lar error will affect the horizontal distance by nearly the 
same amount, whether the ground be level or steep, but will 
affect the vertical height very much more on steep than on 
level ground. It may be further shown that it is necessary 
to keep d tolerably small and s as large as possible, especially 
on steep ground. 

114. Measuring Distances with Gradienter—The gradi-. 

enter is used as a telemeter in measuring horizontal distances 
in two ways: first, by measuring the space on the rod passed 
over by the horizontal cross-hair for a given number of revo¬ 
lutions of the gradienter screw; second, by noticing the 
number of revolutions of the screw required to carry the hori¬ 
zontal cross-hair over a fixed space on the rod. Gradienter 
screws are so made that a single revolution may carry the 




NATURAL SINES AND COSINES. 


275 


Table XI.—natural sines and cosines. 

(From the Smithsonian Tables.) 


NATURAL SINES. 


An¬ 

gle. 

O' 

10' 

20' 

30 ' 

40 ' 

50 ' 

60 ' 

An¬ 

gle. 

Prop. 
Parts 
for 1 ' 

0° 

.0000 00 

.OC29 09 1 

.0058 18 1 

.0087 27 

.0116 35 

.0145 44 

.0174 52 

89° 

2.9 

I 

.0174 52 

.0203 6 

.0232 7 

.0261 8 

.0290 8 

.0319 9 

.0349 O 

88 

2.9 

2 

.0349 0 

.0378 1 

.0407 I 

.0436 2 

•0465 3 

.0494 3 

•0523 4 

87 

2.9 

3 

•0523 4 

•0552 4 

.0581 4 

.0610 5 

.0639 5 

.0668 5 

.0697 6 

86 

2.9 

4 

.0697 6 

.0726 6 

•0755 & 

.0784 6 

.0813 6 

.0842 6 

.0871 6 

85 

2.9 

5 

.0871 6 

.0900 5 

.0929 5 

•0958 5 

.0987 4 

.1016 4 

•1045 3 

84 

2.9 

6 

• T °45 3 

.1074 2 

.1103 I 

.1132 0 

.1160 9 

.1189 8 

.1218 7 

83 

2.9 

7 

.1218 7 

.1247 6 

.1276 4 

.1305 3 

•1334 

•1363 

.1392 

82 

2.9 

8 

.1392 

. 1421 

.1449 

.1478 

• 1507 

.1536 

.1564 

81 

2.9 

9 

.1564 

•1593 

. 1622 

• 1650 

.1679 

. 1708 

.1736 

80 

2.9 

10 

.1736 

•1765 

.1794 

. 1822 

.1851 

. 1880 

. 1908 

79 

2.9 

11 

. 1908 

•1937 

.1965 

.1994 

. 2022 

. 2051 

.2079 

78 

2.9 

12 

. 2079 

.2108 

.2136 

. 2164 

.2193 

. 2221 

.2250 

77 

2.8 

13 

. 2250 

. 2278 

. 2306 

•2334 

.2363 

.2391 

. 2419 

76 

2.8 

14 

.2419 

.2447 

.2476 

.2504 

.2532 

.2560 

.2588 

75 

2.8 

15 

. 2588 

. 2616 

.2644 

. 2672 

. 2700 

. 2728 

.2756 

74 

2.8 

16 

.2756 

. 2784 

.2812 

. 2840 

. 2868 

. 2896 

. 2924 

73 

2.8 

17 

. 2924 

.2952 

.2979 

.3007 

•3035 

. 3062 

.3090 

72 

2.8 

18 

.3090 

.3118 

.3145 

•3173 

•3201 

.3228 

•3256 

7 i 

2.8 

19 

.3256 

.3283 

•3311 

•3338 

•3365 

•3393 

•3420 

70 

2.7 

20 

.3420 

•3448 

•3475 

.3502 

•3529 

•3557 

.3584 

({9 

2.7 

21 

.3584 

.3611 

.3638 

.3665 

.3692 

•3719 

.3746 

68 

2.7 

22 

•3746 

•3773 

.3800 

.3827 

•3854 

.3881 

•3907 

67 

2.7 

23 

• 3907 

•3934 

• 39 61 

•3987 

.4014 

.4041 

.4067 

66 

2.7 

24 

.4067 

.4094 

.4120 

•4147 

•4173 

.4200 

.4226 

65 

2.7 

25 

.4226 

•4253 

•4279 

•4305 

• 433 i 

•4358 

•4384 

04 

2.6 

26 

•4384 

. 4410 

•4436 

.4462 

. 4488 

•4514 

•4540 

63 

2.6 

27 

-4540 

.4566 

• 459 2 

.4617 

•4643 

.4669 

• 4 6 95 

62 

2.6 

28 

• 4 6 95 

•4720 

.4746 

•4772 

•4797 

• 4823 

. 4848 

61 

2.6 

29 

. 4848 

.4874 

•4899 

•4924 

•4950 

•4975 

.5000 

60 

2-5 

30 

.5000 

.5025 

.5050 

•5075 

.5100 

•5125 

•5150 

59 

2.5 

31 

•5150 

• 5175 

. 5200 

•5225 

•5250 

.5275 

•5299 

58 

2.5 

32 

•5299 

.5324 

•5348 

•5373 

•5398 

• 5422 

.5446 

57 

2.5 

33 

• 5446 

• 5471 

• 5495 

•5519 

• 5544 

•5568 

•5592 

56 

2.4 

34 

.5592 

.5616 

. 5640 

.5664 

.5688 

•5712 

•5736 

55 

2.4 

35 

. 5736 

.5760 

•5783 

.5807 

.5831 

•5854 

.5878 

54 

2.4 

36 

. 5878 

.5901 

•5925 

.5948 

•5972 

•5995 

. 6018 

53 

2.3 

37 

.6018 

.6041 

.6065 

.6088 

.6111 

.6134 

•6157 

52 

2.3 

38 

.6157 

.6180 

.6202 

.6225 

.6248 

.6271 

.6293 

5 i 

2.3 

39 

.6293 

.6316 

•6338 

.6361 

•6383 

. 6406 

.6428 

50 

2-3 

40 

. 6428 

.6450 

.6472 

.6494 

•6517 

•6539 

.6561 

49 

2.2 

AT 

.6561 

•6583 

.6604 

. 6626 

.6648 

.6670 

.6691 

48 

2.2 

J.2 

. 6691 

.6713 

• 6734 

.6756 

.6777 

.6799 

. 6820 

47 

2.2 


. 6820 

. 6841 

.6862 

.6884 

.6905 

.6926 

.6947 

46 

2 . I 

44 

.6947 

.6967 

.6988 

1 • 7009 

.7030 

.7050 

.7071 

45 

2. I 


GO' 

50 ' 

40 ' 

• 

30 ' 

| 20' 

10 ' 

0 

A ri¬ 
ffle. 



NATURAL COSINES. 
























































276 


ANGULAR TA CHYMETR 1 


Table XI. —natural sines and cosines. 

NATURAL SINES. 


An- 

gle. 

0 

10' 

20' 

30 ' 

40 ' 

50 ' 

60 ' 

An¬ 

gle. 

li’rop. 
Parts 
for 1 ' 

45 ° 

.7071 

. 7092 

.7112 

•7133 

•7153 

•7173 

•7193 

440 

2.0 

46 

.7193 

.7214 

•7234 

.7254 

.7274 

.7294 

•7314 

43 

2.0 

47 

.7314 

•7333 

•7353 

•7373 

•7392 

.7412 

•7431 

42 

2.0; 

48 

•7431 

•7451 

•7479 

.7490 

•7509 

.7528 

•7547 

4 i 

1.9 

49 

•7547 

.7566 

•7585 

.7604 

.7623 

.7642 

. 7660 

40 

1-9 

50 

. 7660 

.7679 

.7698 

.7716 

•7735 

• 7753 

•7771 

39 

1.9 

5 1 

.7771 

•7790 

. 7808 

. 7826 

• 7844 

. 7862 

. 7880 

38 

1.8 

! 52 

. 7880 

.7898 

.7916 

•7934 

•7951 

.7969 

.7986 

37 

1.8 

53 

.7986 

.8004 

.8021 

.8039 

.8056 

•8073 

. 8090 

36 

i -7 

54 

. 8090 

.8107 

1 .8124 

.8141 

.8158 

•8175 

.8192 

35 

i -7 

r r 

00 

.8192 

. 8208 

.8225 

. 8241 

.8258 

.8274 

. 8290 

34 

1.6 

56 

. S290 

•8307 

•8323 

•8339 

•8355 

•8371 

.8387 

33 

1.6 

57 

.8387 

•8403 

.8418 

•8434 

.8450 

.8465 

. 8480 

32 

1.6 

58 

. 8480 

. 8496 

.8511 

• 8526 

.8542 

•8557 

• 8572 

3 i 

i -5 

59 

.8572 

.8587 

. 8601 

.8616 

.8631 

.8646 

.8660 

30 

i -5 

60 

. 8660 

.8675 

.8689 

.8704 

.8718 

.8732 

.8746 

29 

1.4 

61 

.8746 

. 8760 

•8774 

.8788 

. 8802 

.8816 

. 8829 

28 

1.4 

62 

. 8829 

•8843 

•8857 

.8870 

.8884 

.8897 

.8910 

27 

1 • 4 

63 

. 8qio 

• 8923 

.8936 

•8949 

. 8962 

•8975 

.8988 

26 

x *3 

64 

.89S8 

. 9001 

.9013 

.9026 

.9038 

• 9 ° 5 I 

.9063 

25 

1 3 

65 

.9063 

•9075 

.9088 

. 9100 

.9112 

.9124 

•9135 

24 

1.2 

j 66 

•9135 

.9147 

•9159 

.9171 

. 9182 

.9194 

• 9205 

23 

1.2 

i 67 

63 

• 9205 

.9216 

.9228 

•9239 

.9250 

. 9261 

.9272 

22 

1.1 

.9272 

• 9283 

•9293 

•9304 

•9315 

•9325 

•9336 

21 

1.1 

69 

•9336 

.9346 

•9356 

•9367 

•9377 

•9387 

•9397 

20 

1.0 

70 

•9397 

.9407 

.9417 

.9426 

•9436 

.9446 

•9455 

19 

1.0 

71 

•9455 

.9465 

•9474 

•9483 

.9492 

• 9502 

• 95 ii 

18 

0.9 

72 

• 95 ii 

.9520 

.9528 

•9537 

•9546 

• 9555 

•9563 

17 

0.9 

73 

•9563 

•9572 

.9580 

.9588 

•9596 

■ 9 6 °5 

.9613 

16 

0.8 

74 

.9613 

.9621 

. 9628 

.9636 

.9644 

.9652 

•9659 

15 

0.8 

i 75 

•9659 

. 9667 

.9674 

.9681 

. 9689 

. 9696 

•9703 

14 

0.7 

76 

•9703 

.9710 

•9717 

•9724 

•9730 

•9737 

•9744 

13 

0,7 

77 

•9744 

•9750 

•9757 

•9763 

.9769 

• 9775 

• 978 i 

12 

0.6 

78 

• 97 Si 

-9787 

•9793 

•9799 

.9805 

.9811 

.9816 

II 

0.6 

79 

.9816 

. 9822 

.9827 

•9833 

• 9838 

•9843 

.9848 

IO 

0.5 

SO 

.9848 

•9853 

• 9858 

•9863 

.9868 

.9872 

•9877 

9 

0-5 

81 

.9877 

. 9881 

.9886 

.9890 

.9894 

•9899 

•9903 

8 

0.4 

82 

•9903 

.9907 

.9911 

•9914 

.9918 

.9922 

•9925 

7 

0.4 

83 

•9925 

.9929 

•9932 

•9936 

•9939 

.9942 

•9945 

6 

o -3 

84 

•9945 

•9948 

•9951 

•9954 

•9957 

•9959 

.9962 

5 

°-3 

85 

.9962 

.9964 

.9967 

.9960 

.9971 

•9974 

.9976 

4 

0.2 

86 

.9976 

.9978 

.9980 

.9981 

•9983 

•9985 

.9986 

3 

0.2 

87 

. 9986 

. 9988 

•9989 

.9990 

•9992 

•9993 

•9994 

2 

0.1 

88 

•9994 

•9995 

.9996 

•9997 

•9997 

•9998 

.9998 

1 

0.1 

89 

•9998 

•9999 

•9999 

1.0000 

1.0000 

1.0000 

r.0000 

0 

0.0 


60 ' 

50 ' 

40 ' 

30 ' 

20 ' 

10' 

0 

An¬ 

gle. 



NATURAL COSINES. 


































































NATURAL TANGENTS AND COTANGENTS. 277 


Table XII.—natural tangents and cotangents. 

(From the Smithsonion Tables.) 

NATURAL TANGENTS. 


An¬ 

gle. 

O' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

00 ' 

An¬ 

gle. 

Prop. 
Parts 
for i' 

0 ° 

.0000 0 

.0029 I 

.0058 2 

.0087 3 

.0116 4 ! 

•0145 5 

.0174 6 

80 ° 

2.9 

I 

.0174 6 

.0203 6 

.0232 8 

.0261 9 

.0291 0 

.0320 1 

.0349 2 

88 

2 -9 

2 

.0349 2 

•0378 3 

.0407 5 

.0436 6 

.0465 8 

.0494 9 

.0524 I 

87 

2.9 

3 

.0524 1 

•0553 3 

.0582 4 

.0611 6 

.0640 8 

.0670 0 

.0699 3 

86 

2.9 

j 4 

.0699 3 

.0728 5 

•0757 8 

O 

CO 

q 

.0S16 3 

.0845 6 

.0874 9 

85 

2.9 

.» 

.0S74 9 

.0904 2 

•0933 5 

.0962 9 

.0992 3 

.1021 6 

.1051 O 

SI 

2.9 

6 

.1051 0 

.1080 5 

.1109 9 

.1139 4 

.1168 8 

.1198 3 

.1227 8 

83 

2.9 

7 

.1227 8 

.1257 4 

.1286 9 

•1316 5 

.1346 

.1376 

.1405 

S2 

3 -o 

8 

.1405 

•1435 

.1465 

• 1495 

.1524 

• 1554 

.1584 

8l 

3-0 

9 

.1584 

. 1614 

.1644 

• 1673 

.1703 

•1733 

.1763 

SO 

3 0 

10 

.1763 

•1793 

.1823 

.1853 

.1883 

.1914 

.1944 

71 ) 

3 -o 

11 

.1944 

.1974 

.2004 

.2035 

.2065 

.2095 

.2126 

7 S 

3 -o 

12 

.2126 

.2156 

.2186 

.2217 

.2247 

.2278 

.2309 

77 

3.1 

13 

.2309 

•2339 

.2370 

.2401 

.2432 

.2462 

•2493 

76 

3 -i 

14 

•2493 

.2524 

• 2555 

.2586 

.2617 

.2648 

.2679 

75 

3 -i 

15 

.2679 

.2711 

.2742 

.2773 

.2805 

.2836 

.2867 

71 

3-1 

16 

.2867 

.2899 

.2931 

.2962 

.2994 

.3026 

.3057 

73 

3-2 

17 

.3057 

.3089 

• 3121 

.3153 

.3185 

.3217 

•3249 

72 

3-2 

18 

•3249 

.3281 

.3314 

.3346 

•3378 

• 34 ii 

•3443 

7 i 

3-2 

19 

•3443 

.3476 

.3508 

.3541 

• 3574 

.3607 

.3640 

70 

3-3 

20 

.3640 

• 3673 

.3706 

.3739 

•3772 

.3805 

•3839 

00 

3-3 

21 

•3339 

.3872 

.3906 

•3939 

•3973 

.4006 

.4040 

68 

3-4 

22 

.4040 

.4074 

.4108 

.4142 

.4176 

.4210 

• 4245 

67 

3-4 

23 

.4245 

.4279 

.4314 

.4348 

•4383 

.4417 

• 4452 

66 

3-5 

24 

•4452 

.4487 

.4522 

.4557 

•4592 

.4628 

.4663 

65 

3-5 

25 

.4663 

.4699 

• 4734 

.4770 

.4806 

.4841 

.4877 

01 

3-6 

26 

.4877 

.4913 

• 4950 

.4986 

.5022 

• 5059 

• 5095 

63 

3-6 

27 

.5095 

•5132 

.5169 

.5206 

.5243 

.5280 

•5317 

62 

3-7 

28 

.5317 

•5354 

.5392 

•5430 

.5467 

.5505 

•5543 

61 

3-8 

29 

• 5543 

.5581 

.5619 

.5658 

.5696 

•5735 

•5774 

60 

3-8 

30 

• 5774 

.5812 

.5851 

.5890 

•5930 

• 59 6 9 

.6009 

50 

3-9 

31 

.6009 

.6048 

.6088 

.6128 

.6168 

.6208 

.6249 

5 S 

4.0 

32 

.6249 

.6289 

.6330 

.6371 

.6412 

•6453 

.6494 

57 

4.1 

33 

.6494 

.6536 

.6577 

.6619 

.666r 

.6703 

.6745 

56 

4-2 

34 

.6745 

.6787 

.6830 

.6873 

.6916 

.6959 

.7002 

55 

4-3 

35 

.7002 

.7046 

.7089 

•7133 

• 7177 

.7221 

.7265 

•)1 

4-4 

36 

.7265 

.7310 

• 7355 

.7400 

•7445 

.7490 

.7536 

53 

4-5 

37 

.7536 

.7581 

.7627 

.7673 

.7720 

.7766 

.7813 

52 

4.6 

3 S 

•7813 

.7860 

.7907 

.7954 

.8002 

.8050 

.8098 

5 i 

4-7 

39 

.8098 

.8146 

.8195 

.S243 

.8292 

.8342 

.8391 

50 

4-9 

1 

40 

.8391 

.8441 

.8491 

.8541 

.8591 

.8642 

.8693 

10 

5.0 

41 

.8693 

.8744 

.8796 

.8847 

.8899 

.8952 

.9004 

48 

5-2 

42 

.9004 

• 9057 

.9110 

.9163 

.9217 

.9271 

•9325 

47 

5.4 

43 

•9325 

.9380 

•9435 

.9490 

• 9545 

.9601 

• 9657 

46 

5.5 

44 

• 9^57 

.9713 

.9770 

.9827 

.9884 

•9942 

1.0000 

45 

5 ; 7 


Go' 

50 ' 

40 ' 

30 ' 

20 ' 

10 ' 

0 ' 

An 

ele. 



NATURAL COTANGENTS. 























































278 


A NG ULA R TA CH YME TR Y. 


Table XII.—natural tangents and cotangents. 

NATURAL TANGENTS. 


An¬ 

gle. 

0' 

10' 

20' 

30 ' 

40 ' 

50 ' 

60 ' 

A n- 
gle. 

Prop. 
Parts 
for P 

45° 

1.0000 

I.OO58 

1.0117 

1.0176 

I.0235 

1.0295 

1-0355 

440 

5-9 

46 

1-0355 

1.0416 

I.0477 

I.0538 

I.0599 

1.0661 

I.0724 

43 

6.1 

47 

1.0724 

1.0786 

I.0850 

I.0913 

I.0977 

1.1041 

1.1106 

42 

6.4 

48 

1.1106 

I.II 7 I 

I-I 237 

I .1303 

1.1369 

1.1436 

1 .1504 

4 i 

6.6 

49 

1.1504 

I -1571 

I.1640 

1.1708 

1.1778 

1.1847 

I.191S 

40 

6.9 

50 

1.1918 

I.1988 

1.2059 

1.2131 

I.2203 

1.2276 

I.2349 

39 

7-2 

51 

1.2349 

I.2423 

I.2497 

I.2572 

1.2647 

1.2723 

I.2799 

38 

7-5 

52 

1.2799 

I.2876 

I.2954 

I.3032 

1.31U 

1.3190 

I.3270 

37 

7-9 

33 

1.3270 

i- 335 i 

1-3432 

I- 35 I 4 

1-3597 

1.3680 

I .3764 

36 

8.2 

5 hf 

1-3764 

1.3848 

1-3934 

1.4019 

1.4106 

1-4193 

I 4281 

35 

8.6 

00 

1.4281 

1-4370 

1.4460 

1-4550 

I.4641 

1-4733 

I.4826 

34 

9 1 

56 

1.4826 

I-49I9 

1-5013 

I.5108 

1.5204 

1. 53 oi 

1-5399 

33 

9.6 

57 

1-5399 

1-5497 

1-5597 

I-5697 

I -5798 

1.5900 

I.6003 

32 

10.1 

58 

1.6003 

1.6107 

1.6212 

1.6319 

1.6426 

1-6534 

1.6643 

3 i 

10.7 

59 

1.6643 

1.6753 

1.6864 

I.6977 

I.7090 

1.7205 

1.7321 

30 

11 • 3 

60 

1.7321 

1-7437 

1.7556 

I.7675 

I.7796 

I-79I7 

I.8040 

29 

12.0 

61 

1.8040 

1.8165 

1.8291 

I.8418 

1.8546 

1.8676 

1.8807 

28 

12.8 

62 

1.8807 

1.8940 

1.9074 

I.9210 

1-9347 

1.9486 

I.9626 

27 

13.6 

63 

1.9626 

1.9768 

1.9912 

2.0057 

2.0204 

2.0353 

2.0503 

26 

14.6 

64 

2.0503 

2.0655 

2 0809 

2.0965 

2.1123 

2.1283 

2.1445 

25 

15-7 

65 

2.1445 

2.1609 

2.1775 

2.1943 

2.2113 

2.2286 

2.2460 

24 

16.9 

66 

2.2460 

2.2637 

2.2817 

2.2998 

2.3183 

2.3369 

2-3559 

23 

18.3 

67 

2-3559 

2-3750 

2-3945 

2.4142 

2.4342 

2-4545 

2.4751 

22 

19-9 

68 

2.4751 

2.4960 

2.5172 

2.5386 

2.5605 

2.5826 

2.6051 

21 

21.7 

69 

2.6051 

2.6279 

2.6511 

2.6746 

2.69S5 

2.7228 

2-7475 

20 

23-7 

70 

2-7475 

2.7725 

2.7980 

2.8239 

2.8502 

2.8770 

2.9042 

19 


7i 

2.9042 

2.9319 

2.9600 

2.9887 

3.OI78 

3-0475 

3-0777 

18 


72 

3.0777 

3.1084 

3-1397 

3 .I 7 I 6 

3.2041 

3 2371 

3.2709 

17 


73 

3.2709 

3-3052 

3.3402 

3-3759 

3.4124 

3-4495 

3-4874 

16 


74 

3-4874 

3 - 526 i 

3-5656 

3.6059 

3.6470 

3-6891 

3-7321 

15 


75 

3-7321 

3 . 776 o 

3.8208 

3.8667 

3 - 9 I 36 

3-9617 

4.0108 

14 


76 

4.0108 

4.0611 

4.1126 

4.1653 

4-2193 

4.2747 

4-3315 

13 


77 

4-3315 

4-3897 

4.4494 

4-5107 

4-5736 

4.6382 

4.7046 

12 


73 

4.7046 

4.7729 

4-8430 

4 - 9 I 52 

4.9894 

5.0658 

5 1446 

11 


79 

5.1446 

5.2257 

5-3093 

5-3955 

5-4845 

5-5764 

5.6713 

10 


80 

5-6713 

5-7694 

5.8708 

5-9758 

6.0844 

6.1970 

6.3138 

9 


81 

6.3138 

6.4348 

6.5606 

6.6912 

6.8269 

6.9682 

7-1154 

8 


82 

7 -II 54 

7.2687 

7.4287 

7-5958 

7.7704 

7-9530 

8.1443 

7 


83 

8.1443 

8.3450 

8-5555 

8.7769 

9.OO98 

9-2553 

9-5144 

6 


84 

9 - 5 M 4 

9.7882 

10.0780 

10.3S54 

IO.7II9 

11.0594 

ii- 53 °i 

5 


85 

11.4301 

11.8262 

12.2505 

12.7062 

I3.I969 

13.7267 

14 3007 

4 


86 

14.3007 

14.9244 

15.6048 

16.3499 

17.1693 

18.0750 

19.0811 

3 


87 

19.0811 

20.2056 

21.4704 

22.9038 

24 - 54 I 8 

26.4316 

28.6163 

2 


88 

28.6363 

31.2416 

34.3678 

38.1885 

42.9641 

49.1039 

57-2900 

i 


89 

57.2900 

68.7501 

85.9398 

114-5887 

I 7 I -8854 

343-7737 

0 © 

0 



60 ' 

60 ' 

40 ' 

30 ' 

20' 

10' 

0' 

An- 



ffle 


NATURAL COTANGENTS. 





































MEASURING DISTANCES WITH GRADIENTER. 2 J() 

hair over either one foot or two feet on the stadia-rod at a 
distance of ioo feet. Prof. Ira O. Baker gives the following 
formula for measuring the distance by means of gradienter 
used in either of the above ways: 

D — ioo i, .(19) 

in which D — horizontal distance in feet between instrument 
and rod, and i = intercept on the rod for one revolution of 
the gradienter screw. 

This fundamental equation corresponds closely with that 
for the stadia and applies to a rod held perpendicular to the 
line of sight. In working on slopes, however, the rod will 
for convenience be held vertically, when the line of sight will 
be inclined to the rod, and the formula for this case then 
becomes 

D — z’(ioo cos a — sin . (20) 

in which a is the angle between the lower visual ray and the 
horizontal. The above gives the distance directly observed 
on the lower visual ray, and from it we derive 

D — /(ioo cos 3 a — sin 2 <*). . . . (21) 

P'or a constant intercept on the rod, the formula deduced 
by Prof. Baker is 

IOO 

D = — 5 ,.(22) 

n v ' 

in which 5 is the distance between the fixed targets and n = 
number of revolutions required to move the line of sight over 
the constant intercept at the distance D. The horizontal 
distance then becomes 

100 cos’ a 

d =-o, 



n 






28 o 


ANG ULAR TA CHYME TR Y. 


115. Wagner-Fennel Tachymeter. —By means of an in¬ 
strument made in Germany and known by the above name, 
surveys to be completed on large scale and with great 
detail may be conducted more rapidly than with ordinary 
transit and stadia. There are tzvo forms of this instrument , 
both of which are so arranged that the horizontal distance and 
absolute height of the point to be determined are read direct 
from the instrument after the simple pointing and some inter¬ 
mediate manipulation, without moving the telescope or mak¬ 
ing any computations. The first of these instruments (Fig. 
83) corresponds to a transit- and the second to an alidade. 
The latter called a tachygraphometer, for use with the plane- 
table, will probably be of service on large-scale surveys in which 
the elevations of numerous points are to be determined. 
With this instrument the positions and elevations of the 
points can be plotted on the plane-table in the field with 
great precision and facility'and without the danger of omit¬ 
ting details of form. Both instruments are fully described 
and figured in Appendix 16, Report of U. S. Coast and Geo¬ 
detic Survey for 1891. 

The field observations with this class of instrument consist in 
determining, first, the distance on the slope, and then the hori¬ 
zontal and vertical angle without taking account as a separate 
observation of the space subtended on the rod from which 
the inclined distance is determined. From these data the 
azimuth of the desired point is determined, then the reduced 
distance and relative height from the point of observation. 
These operations are all mechanical and graphical, calling for 
no computations whatever. The manipulation of the instru¬ 
ment is simple and, with practice, rapid. It is adjusted to 
the occupied station, and the height of the latter set upon 
the scale of heights. The determination of separate points is 
then proceeded with in the following order: 

The rodman sets his rod at the desired point; the instru- 
mentman brings the middle wire upon the zero point of the 


IVA GNER-FENNEL TA CH YME TER 


28l 


rod, reads the space on the rod intercepted by the distance- 
wires, and records in his note-book the corresponding inclined 



I 1 '.' | 1 r ipfcJ 
1 * n : Hm 1 '] 




Otto Fennel,.Cassei 


iMMIHiHlll! 


Si®! 




F 1G . 83.— Wagner-Fennel Theodolite Tachymeter. 

distance, T hese he sets off upon the rule parallel to the line 
of sight, pressing the projection angle against the vernier of 











































282 


A NG ULA R TA CH YME TR V. 


heights, and from the latter reads the required height, and 
from the horizontal rule the reduced distance of the observed 
point. Lastly, the horizontal angle is read with the tachym- 
eter or the tachygeometer and plane-table. The first por¬ 
tion of the operations with the two instruments are the same, 
excepting that with the latter the horizontal projection of the 
observed point is pricked upon the drawing-paper in its cor¬ 
rect position by pressing a needle suitably arranged against 
the horizontal scale instead of recording the same in a note¬ 
book. These various operations take from one and a half to 
two minutes. This instrument can be used satisfactorily for 
distances of a thousand feet, making the central base of the 
instrument two thousand feet. 

116. Range-finding. —Range-finders are instruments which 
are primarily intended for military use in determining the dis¬ 
tances or ranges of objects as an aid to directing artillery fire. 
There are many forms of this instrument, some of which are 
exceedingly large and elaborate and are fixed permanently, 
as those employed on seacoast fortifications. Of the smaller 
and more portable forms used with light horse-batteries, that 
known as the Weldon range-finder, from its inventor, has 
given the most satisfactory results in actual practice. 

Range-finding may also be done with the plane-table, on 
the same scale when this is sufficiently large; on a provision¬ 
ally large scale when the scale of the map is small. The light 
plane-table as a traverse or triangulation instrument, in con¬ 
nection with its use as a range-finder for distances, and with 
a vertical-angle sight-alidade for elevations, furnishes a most 
satisfactory tachymeter, both for filling in details on large- 
scale maps, and for carrying on rough geographic or explora¬ 
tory surveys. 

The range-finder furnishes a satisfactory rough telemetric 
method of obtaining a fairly accurate measure of inaccessible 
distances. Pacing or time-sketching may be depended upon 
where the surveyor may travel, but over rough country or for 


SURVEYING WITH RANGE-FINDER. 283 

determining the positions of points on either side of a traversed 
route the range-finder is unequalled except by intersection 
methods, and the latter can only be employed where the in¬ 
strument may be set up and angles taken by which to obtain 
intersections. The range-finder is most useful in military 
sketching and in route surveying. The more important ad¬ 
vantages which it has are in enabling the surveyor to fix the 
position of a number of points which lie within the limits of 
his vision from one point. In ordinary surveying, to measure 
the distance of an object 5000 feet or more away with any 
degree of accuracy by intersections would require a base of 
at least 2000 feet in length or, better still, according to theo¬ 
retical methods, 5000 feet in length; yet with the range¬ 
finder the same distance can be measured with comparative 
accuracy from a base but 100 feet in length. 

116 a. Estimating Distances. — Commander William H. 
Beehler, U. S. N., adopted a simple method for estimating 
distances by getting two lines of sight on a distant object, one 
with the right and the other with the left eye. The observer 
extends his right arm with the forefinger pointing to the object, 
and sights with his right eye along the arnC Then, holding the 
arm and finger rigid, he doses his right eye and sights with his 
left eye over the point of The forefinger. He finds that the 
second sight-line will point to the right of the distant object a 
distance equal to one-tenth of the distance of the object from 
the observer. The lateral distance to the right must be estim¬ 
ated; and the accuracy of the estimate depends upon practice 
in observing distant objects of known dimensions. The lateral 
error is, of course, magnified ten times when applied to the 
horizontal distance between the observer and the distant object. 

As men are not uniform in size, and the distance between 
the eyes is not always the same, the ratio must be determined 
in each case. But the proportion of 10 to 1 should be retained 
as the most convenient, even if the point of intersection for the 


284 


ANGULAR TACHYMETRY. 


left eye sight has to be prolonged by the use of a lead pencil, or 
other object. 

In principle this method is similar to the old topographer’s 
manner of estimating distances by the use of string and a grad¬ 
uated lead-pencil. In this latter method a knotted string is 
used attached to the pencil at one end, and the knot at the 
other end held in the teeth. In this case, the pencil—which is 
held vertically at arm’s length, has been previously graduated 
by observing familiar objects at known distances; these objects 
being a man on foot or on horseback, the ordinary height of a 
fence, or any common object that may be selected. The 
graduations on the pencil then correspond to known horizontal 
distances. To use this method, the pencil is held out at the 
known distance from the right eye—as fixed by the knotted 
string, and the line of sight to the top of the distant object runs 
over the top of the pencil, holding the arm rigid, the thumb is 
then slipped down until the second sight-line strikes the foot of 
the distant object; the position of the thumb then fixes the 
horizontal distance. The difficulty in using this method with 
any assurance of accuracy is the difficulty of always finding an 
object of known altitude to sight at. 

117. Surveying with Range-finder. —The range-finder 
can be most successfully used in three ways. First, in geo¬ 
graphic or topographic surveying, while occupying a plane- 
table or triangulation station the position of which is known, 
and which is surrounded by a few locations. The remainder 
of the country can be sketched by locating with the range¬ 
finder numerous minor points which will so control the 
sketching as to permit of a greater amount being done from 
one station than could be accomplished by other methods. 
Even in comparatively detailed topographic work the range¬ 
finder may be thus used in place of the stadia, for with the 
latter rodmen would have to be sent to the points the posi¬ 
tions of which are to be determined, while with the range- 


TRAVERSING WITH RANGE-FINDER. 285 

finder it is but necessary to obtain a definite object to which 
to sight. 

The second method of using the range-finder is in traverse 
or route surveying , where the positions of points on either 
side of the route can be determined by the range-finder more 
rapidly than by setting up an intersection instrument, and 
the country thus controlled by points ranged on either side 
can be rapidly sketched in. 

The third method of using the range-finder is by employ¬ 
ing it to measure distances along the route traversed when 
the latter is especially irregular or winding. Thus the trav¬ 
eled route may be measured by ordinary means only by 
going over the ground along which the line of sight is taken ; 
but with the range-finder, as with the stadia or other tele¬ 
metric instrument, though the road twist and turn and wind 
about in a ravine, canyon, or over tortuous country, it is 
unnecessary to measure the route traveled. It suffices to 
range-in some object in the line of travel and plat the same, 
when the surveyor may pursue any route he chooses to reach 
that object without the necessity of measuring the distance 
as he progresses, the same having already been obtained by 
the range-finder. 

The extreme adaptability of the range-finder may be 
realized when it is known that a base can be accurately meas¬ 
ured between two points selected as convenient stations for 
its use without taking into consideration the irregularities of 
the ground between them. In other words, it is not neces¬ 
sary to directly measure by pacing or taping the base used 
with the range-finder, but it is perfectly feasible to take the 
platted distance between two inaccessible points, as deter¬ 
mined by a good range-finder, from a third point whence the 
two in question are both visible. 

118. Traversing with Range-finder.—The most satis¬ 
factory method of using the range-finder in traversing or route¬ 
surveying is that described by Capt. Willoughby Verner, R.E., 
in which he used a combination of range-finder, compass, and 


286 


ANGULAR TACHYMETRY. 





Fig. 84. —Reconnaissance Sketch-map with Cavalry-board and Range¬ 
finder. After Capt. Willoughby Verner. 





















































WELD ON A’A NGE-FINDER. 


287 


intersection which enabled him to sketch a considerable dis¬ 
tance to either side of the route traversed. The directions 
were taken with a cavalry sketch-board (Art. 64) mounted on 
a tripod, and distances were observed with the range-finder. 
At the starting-point (Fig. 84) a round of directions were 
drawn on the board, and the ranges of a number of the more 
prominent objects were taken with the range-finder and their 
positions marked on the sketch. He then rode rapidly to 
high ground 3175 yards distant, the direction and range of 
which had been plotted from the first point. Arrived there, 
the first thing was to find a conspicuous point in the direction 
to be traveled, which was again ranged in and plotted on the 
board, its distance being 1980 yards. 

The board being mounted on a tripod and oriented by the 
needle, intersections were taken on a number of points pre¬ 
viously indicated by direction lines, while new direction lines 
were plotted to various objects, a few of which were again 
ranged-in, and this process was continued. Its chief advan¬ 
tages were that the surveyor was able to ride rapidly over the 
ground, along the most accessible route, from one point to 
another, and to locate a number of points in every direction, 
some by intersection, others by ranging. Sometimes the 
range in the direction of the route of travel is obstructed by 
trees or other objects; when it is possible to sight in that di¬ 
rection on the sketch-board, measure the distance by pacing 
or otherwise until the obstacle is passed, and then resume the 
range-finding. 

119. Weldon Range-finder —This instrument consists of 
three prisms accurately ground to the following angles: first, 
90 degrees; second, 88 degrees 51 minutes 15 seconds; 
third, 74 degrees 15 minutes 53 seconds. The distance or 
range of an object , O (Fig. 85), from an observer is obtained 
by observing the angles OAD and OB A at the base of a 
right-angled triangle, ABO , the measured base, AB } of which 
bears the ratio 1 to 50 of the distance or range AO when the 
first or 90-degree prism and the second or 88-degree prism 


288 


A NG ULAR TA CH YME TR Y. 


are used at either end of the base. A more accurate deter¬ 
mination of the range may be obtained by use of the second 
prism only when the measured base is I : 25 of the distance or 
range AO (Fig. 86), in which case the angles of an isosceles 


triangle at ABO and A CO at eit 
measured. Finally, the Weldon 
measuring rapidly a base AB or 


o 



Fig. 85. —Range-finding with 
a Direction-point D . 


her end of the fixed base are 
range-finder may be used for 
BC by using the third prism 


o 



Fig. 86.—Range-finding with 
out Direction-point. 


of 74 + degrees (Fig. 87), but this is merely as a convenience 
and not as a necessity except under very unusual circumstances. 

In taking a range choose a good direction-point, D y 
(Fig. 85), or else put in a ranging rod at D, making its reflec- 



Fig. 87. —Measuring Long Base with Range-finder. 








ACCURACY AND DIFFICULTIES OF RANGE FINDING. 289 

tion coincide with the object by means of the 90° prism; then, 
using the 38 + prism, retire along the line AB, leaving a mark 
at A to keep yourself in line, and when the 88° -j- prism shows 
a coincidence of D with O, B is reached. AB is then meas¬ 
ured either by pacing or, more accurately, by the tape, and 
multiplied by 50, the product being the range of O from A. 

The Weldon range-finder is manufactured in two forms : 1, 
as a small watch-shaped affair about 2 inches in diameter; and, 

2, in a semi-cylindrical case about ij by 2 \ 
inches. (Fig. 88.) The latter, which is the 
most serviceable of the two, has the three 
prisms arranged one above the other, and is 
used by holding it directly in front of the eye, 
grasping the projecting case as a handle, and 
with the first or 90° prism uppermost. The 
apex of this prism is then held in a direct 
line or range with one of the objects sighted, 
and is superimposed over the edge of the 
metal back. This object is then looked at by 
direct vision through the open space above or 
below the prism, and is viewed simultaneously 
with the second object reflected at right angles 
in the prism. 

The reflected object appears directly above 
or below that seen by direct vision, and forms 
with the eye of the observer the angle to which the prism is 
cut. Considerable practice is required to readily determine in 
the prism the reflected object, but by holding the instrument 
quite close to the eye a large field is obtained and a slight 
movement of the head to either side is all that is required to 
bring a fresh field into view. When the reflected image has 
been obtained it can be made to move up or down by slightly 
tilting the instrument so as to make the reflection coincide 
with the object selected in front of the observer for direct 
viewing. In order to get a correct angle the object should 
be kept upright and the reflected horizon as level as it is in 
nature, since any inclination affects the angle. 



Fig. 88.—Weldon 
Range-finder. 





















































































290 


A NG ULA R T.A CH YME 7 R V. 


If in range-finding a good natural direction-point cannot 
be found, a flag or other mark may be placed to get a direc¬ 
tion-point, and its distance from the observer is dependent on 
the distance of the object the range of which is required. 
Thus for a range of 3000 feet the direction-point may be 150 
feet away, but for a mile to two miles the marker should 
certainly not be nearer than 200 to 300 feet. In fact, the 
further the direction-point from the observer the more accu¬ 
rate is the measurement of the range, other conditions being 
equal. If an assistant accompanies the observer, he may be 
used as a direction-point, when very little time will be lost 
in finding one for a long range. 

120. Accuracy and Difficulties of Range-finding. —The 
accuracy obtained with the Weldon range-finder is remarkable 
considering its crudity as a surveying instrument. In tests 
for accuracy made at the Infantry and Cavalry School at Fort 
Leavenworth, Kansas, distances of 2000 and 3500 feet were 
determined within 2.5^ error in every case, and an average 
for a large number of observations and for distances of 2000 
to 12,000 feet, measured by enlisted soldiers unaccustomed 
to the use of the instrument, was 2.43$. 

The chief difficulty in the use of the instrument is one in¬ 
herent in any prismatic instrument; namely, that the object 
the range of which is desired is often hidden from the further 
end of the base by an intervening tree, knoll, or other obsta¬ 
cle, so that, except under very favorable circumstances, several 
trials are necessary in order to get a range, whereas this even 
is sometimes impossible. Other objections to this apparently 
simple process are the difficulty of obtaining a definite mark 
at right angles to the object of reflection when employing 
the base of 1 : 50, and the difficulty of always finding ground 
suitable for the measurement of a base as regards view, gen¬ 
eral configuration, and space. Again, considerable practice is 
necessary in order to obtain reliable results every time, and to 
attain facility in the selection of suitable range-points. And 


RANGE-FINDING WITH PLANE-TABLE. 


29! 


.:c 


finally there is the difficulty, soon overcome with practice, of 
learning to recognize the reflected image, and of producing 
the coincidence of this and the direct view of the range-point. 

121. Range-finding with Plane-table.—Range-finding 
may be performed with a plane-table as satisfactorily as with 
the prismatic range-finder for all the purposes of ordinary 
surveys. The plane-table would, of course, not serve as a 
satisfactory range-finder for military purposes, because it 
offers too large a mark and is not sufficiently portable for 
ordinary military reconnaissance. 

While the plane-table may be satisfactorily employed as a 
range-finder in cases of map-making similar to that described 
in Article 118, its especial adaptability appears 
to be in connection with the determination of 
positions and elevations of unimportant points 
near the route of travel of the topographer 
who is sketching small-scale maps—assuming 
the topographer to be traveling over a road 
previously traversed and adjusted, and sketch¬ 
ing in the topography on either side (Arts. 13 
and 17), and that he finds a point C (Fig. 89), 
either a house in a field a mile or less distant, 
or a summit which has not been previously 
located, and the position and elevation of which 
are essential in order that he may properly 
sketch his surroundings. 

Let him set up his plane-table at a\ then 
orienting by a backsight down the road, if a 
sufficiently long and straight one can be had, 
or by some point x already located and visible 
from the position a, draw a line in the direction 
of C. Now sighting in the direction b } whence 

Range-finding q can a j so k e seerii let him draw the line ab 
with Plane- 

and measure off the base with a tape, or by 


X 

f 


c< 


a b 

Fig. 89. 


TABLE. 


carefully pacing ab say 100 feet. This should then be platted 








291a 


ANGULAR TACHYMETRY. 


on the plane-table on a scale ten times as great as the scale 
of his map. Now removing his plane-table to b and orienting 
on a , let him draw a direction line to C which will approxi¬ 
mately locate it by its intersection with the line from a. The 
angle is necessarily so acute that the actual position of C is 
indefinite, but the distance aC may be scaled off, and this 
divided by iowill reduce it to the scale of the map. Platted 
on the line aC , it will give the location C' so closely, because 
of the great reduction in scale, as to fix the position of the 
point well within the map scale. 

The chief precautions to be taken in this mode of location 
are that the base ab shall not be too small, a ratio of I to 25 
being a very good one and 1 to 50 less satisfactory. Accord¬ 
ingly, with a measured distance ab of 100 feet, a point 2500 
feet distant could be quite accurately platted to a scale 10 
times as great as that of the map. The topographer must take 
especial care in range-finding by this method to set his instru¬ 
ment exactly over the points a and b in order that his orienta¬ 
tion may be accurate. Point a on the plane-table board must 
be plumbed over a stake or other mark, and b likewise must be 
plumbed over the mark sighted at; moreover the backsight 
from b to a must be exactly at the mark a. 

121a. Automatic Surveying Instruments. —Many attempts 
have been made at automatically tracing a plan of routes over which 
self-recording surveying instruments are transported, or of pro¬ 
curing profiles of the same. All of these instruments have, how¬ 
ever, failed to attain any importance in surveying practice of 
even minor accuracy. The only attempt at perfecting an auto¬ 
matic surveying instrument which has even approached success, 
or which has been patented and placed on sale is Ferguson’s 
pedograph. This is, as its name describes, a graphic route 
tracer in which distances are paced. It will map an itinerary 
of moderate length, the extremities of which are known, by in¬ 
terpolating between these. 


AUTOMATIC SURVEYING INSTRUMENTS. 291 b 

The pedograph consists of a U-shaped box, 12 inches in 
diameter by 3 inches in thickness, and hung from the shoulders 
by straps. In it are suspended two frames, on one of which 
is stretched drawing paper, while the other contains a piece of 
ground glass. Between these is a complicated mechanism called 
a recorder, from which is suspended a heavy pendulum which 
causes a hole to be punched in the paper with each step. By 
means of pulleys and cords attached to a compass in the top 
of the box the recorder is caused to change its direction with 
each change in direction of the pedestrian. 


CHAPTER XIV. 


PHOTOGRAPHIC SURVEYING. 

122. Photo-surveying.—The camera has recently come 
into limited favor as a topographic surveying instrument. 
Its first extended use was in Italy, where it was employed 
chiefly in making perspective views of buildings for the pur¬ 
pose of constructing therefrom their elevations and ground- 
plans, for architectural and military purposes, and this form 
of photo-surveying has been styled photogrammetry. As a 
result the word photo-topography has been recently adopted 
as applying to the survey of the terrane by means of the 
camera. Photo-surveying methods have been employed to 
a minor extent in India, France and Italy, and almost ex¬ 
clusively in the Dominion of Canada, in the making of 
topographic surveys. 

123. Photo-surveying and Plane-table Surveying Com¬ 
pared.—A careful study of the method and results of pho¬ 
tographic surveying leads to the following conclusions: 

Photo-surveying consists ultimately in constructing a topo¬ 
graphic map in office from photographs of the terrane in 
conjunction with angular measures taken by the camera. 
Necessarily the draftsman who does not see the country 
cannot make as detailed and accurate a map of it from 
photographs as the topographer could make while viewing 
the country itself from which the photograph had been 
made. It seems, therefore, fair to assert that a map made 
from photographs and constructed in the office on a drawing- 
board, much on the same principle as a map is made on a 

292 


PHO TO-SUP VE YING. 


293 



Fig. 90.—Photograph by Canadian Survey and used in Map Construction. 






































PHOTOGRAPHIC AND PLANE-TABLE SURVEYING. 295 

plane-table board in the field, is less accurate and less satis¬ 
factory than the latter. 

On the other hand, the use of the plane-table requires the 
expenditure of some time in the field in constructing the 
map, while the expenses of a large party organization are 
running on. Considerable outlay is saved in photo-surveying 
by drafting the map in office at the expense of only the indi¬ 
vidual draftsman; moreover, under advantageous conditions 
of light, photo-surveying field operations can be conducted 
more rapidly than plane-table surveys. 

Finally, photo-surveying methods can be employed only 
in mapping a limited class of topographic forms, such as bold 
and open mountainous country, and then only on generalized 
geographic scales. For in highly eroded and detailed topog¬ 
raphy it would be necessary to occupy a multitude of camera 
stations that all the forms might be recorded in photographs. 
In wooded regions, and on plains or plateaus it is impossible 
to use photo-surveying methods. With the plane-table it 
is possible to supplement the facts mapped from the occupied 
stations by any amount of traverse surveying. 

The ultimate conclusion is that a fair map can be made by 
photo-topographic methods, under favorable conditions, more 
rapidly in the field and at less cost than a good map can be 
made on the same scale by plane-table methods. On the 
other hand, where it is desirable to make a first-class topo¬ 
graphic map on a given scale, the best results will be obtained 
with the plane-table under most conditions of atmosphere. 
For it must be borne in mind that when surveying by- 
trigonometric methods, where the topographer leaves camp 
and ascends a mountain to make a plane-table station or 
photographic station, he will under ordinary circumstances, 
succeed in making but one or two stations at most in a day, 
where the scale is of geographic proportions. 

In the average atmospheric conditions met with in the 
United States the topographer'will therefore accomplish as 


2g6 PHOTOGRAPHIC SURVEYING. 

much in a day with the plane-table as with the camera, while 
the resulting map will be decidedly superior. Again, under 
such atmospheric conditions as exist in western British 
America and in Alaska, where the higher summits are covered 
with cloud and mist during the greater portion of the day or 
for several days, and when the occasional glimpses that may 
be had of surrounding country are accompanied by a clear 
and bright sunshine, the topography can be procured by 
photo-topographic methods, completing in an hour of clear 
weather the work necessary to be done at one station, which 
would require the better part of a day by plane-table 
methods. Therefore it is probable that photo-topographic 
methods are cheaper and more rapid than plane-table methods 
and furnish a much more practicable and economic mode of 
making geographic surveys under such conditions. Mr. E. 
Deville, Surveyor-General of Canada, estimates the cost of 
plane-table surveying in western British America, as com¬ 
pared with photo-topographic surveying, as 3 to 1. 

124. Principles of Photo-topography. —The practice of 
photo-topography requires a thorough knowledge of descrip¬ 
tive geometry and perspective. The camera is specially 
prepared, resting on a horizontal plate divided like the circle 
of a transit instrument and read with verniers, and having 
attached to its side or on top a small telescope, with vertical 
arc for the measurement of angles. There are a vertical 
and a horizontal cross-hair in the focal plane of the camera, 
and it is fitted with a magnetic needle inside of the box, 
and a scale, so placed that, when the exposure is made, 
the magnetic declination, the scale, as well as the intersec¬ 
tion of the cross-hairs, are all photographed on the plate 
containing the view (Fig. 91). If the instrument has been 
carefully leveled, the horizontal cross-hair becomes the hori¬ 
zon line, and the vertical cross-hair the center zero line, to 
which angular measures are referred in the office computa¬ 
tions. A group of views are taken at each station, abutting 


PRINCIPLES OF PHOTO-TOPOGRAPHY. 


297 


one against the other, and the angular distance between each 
is noted by the reading of the horizontal plate of the camera, 
horizontal angles being also read by a small theodolite or by 
the camera to the more prominent peaks. 

The objects represented in perspective are of an irregular 
ehape and at various distances from the camera. If the 



Fig. 91.—Bridges-Lee Photo-theodolite. 

picture ^r image of the object is a true perspective in a 
plane, it is possible to construct therefrom a geometric pro¬ 
jection of the object in a plane at right angles to the picture 
plane. This, providing the distance and the relative position 
of the point of view be known with reference to the picture 
plane, and providing views have been taken from a sufficient 

























298 


PHO TOGRA PHIC SUR VE YING. 


number of stations to surround the irregularly formed objects 
'viewed. Photo-topography is , therefore, the art of recon¬ 
structing geometrically horizontal projections from perspec¬ 
tive views. The process of this reconstruction consists in 
platting the skeleton triangulation as obtained by angular 
measures with theodolite or the horizontal circle of the 
camera. The photo-topographic survey should be preferably 
preceded by a primary triangulation. Then, with several 
stations platted, the view from each of them of a given portion 
•of the terrane may be projected on the plane of the map, and 
intersections be platted for each salient point seen in perspec¬ 
tive. 

125. Camera and Plates. —There are a number of forms 
of photo-topographic cameras, among the more complete 
and satisfactory of which are those employed by the Cana¬ 
dian Topographic Survey, the Italian instruments, and the 
Bridges-Lee (Fig. 91) instruments. The apparatus is packed 
in several small cases for easy transportation in the most 
inaccessible country, the tripod, camera, and plates making 
separate packages. The equipment of a photo-topographic 
party in the Canadian surveys consists of a transit theodolite 
and two cameras. These cameras are rectangular boxes of 
metal, open at one end and provided in addition to the lens 
’with two sets of cross-levels, read through openings in the 
•outer mahogany box; the plate-holders are made for single 
plates and are inserted in a frame which can be moved for¬ 
wards and backwards by means of adjusting-screws. The 
.camera rests on a triangular base with leveling foot-screws, 
similar to those of the transit instrument, so that both may 
be used on the same tripod. 

The surveyor first adjusts the transit and measures the 
azimuths and vertical angles to triangulation points and to 
the camera stations, recording the same. The camera is then 
mounted on the tripod, leveled, and the plate-holder inserted, 
and its number is noted, as is also the approximate direction 


FIELD-WORK OF PHOTO-TOPOGRAPHIC SURVEY, 299 

of the view by means of lines drawn on the outer case of the 
camera-box; the topographer then revolves the box until these 
lines show that the camera is properly pointed. Then, by 
looking at the lines on the side of the camera-box, he notes 
whether the-view is in the correct vertical plane. Exposure is 
then made, and the camera sighted for the next view. 

126. Field-work of a Photo-topographic Survey. — In 
the field-work of a photo-topographic survey the primary 
triangulation is first executed by ordinary methods, and 
secondary triangulation is executed during the progress of 
the photo-topographic survey. The object of the secondary 
triangulation being to fix the camera stations, the summits 
located in the secondary triangulation are selected for this 
purpose only, all topographic details of the plat being drawn 
from the photographs made at the camera stations. The 
positions of the camera stations may be fixed either by angles 
from them or by angles from primary triangulation points or 
both, and as it is easier and more accurate to plat the camera 
stations by means of angles taken from the primary triangu¬ 
lation points, the camera stations should, if possible, be 
occupied before the triangulation summits. 

In selecting camera stations it must be borne in mind that 
views taken from a great altitude and overlooking a large 
expanse of country are desirable chiefly as aids in the expan¬ 
sion of the triangulation, while those taken from low altitudes 
are of the greatest service in drawing in details of topography, 
especially in valleys and lowlands. Difficulty is frequently 
experienced in obtaining two views which will furnish inter¬ 
sections over a certain portion of the terrane, in which case 
in very rugged country the method of vertical intersection 
may be employed, views being taken from different altitudes. 
Such a process can of necessity be employed only when the 
differences in elevation are great and the points to be deter¬ 
mined not distant. 

The greatest difficulties in photo-topography are encoun- 


300 


P HO TO G KA PH 1C S UR VE Y1NG. 


tered in bad lights, which must necessarily be met in making 
panoramic views; for while the camera will have the lights in 
the right direction for viewing one way, in taking views in 
the opposite direction the lights will be unfavorable (Art, 
389). Moreover, views taken of the same object or por¬ 
tions of the terrane at different times of the day have the 
shades cast in different manners, so that it becomes difficult 
to identify the topographic detail or even salient points. If 
the number of photographs taken is large enough to cover 
the ground completely, the identificaion of points even under 
different lighting offers no serious obstacle. 

In making exposures two or three points in each view 
must be observed with the altazimuth on the camera or with 
a theodolite, so as to obtain horizontal and vertical angles 
between them, and this aids in the orientation of the view and 
in platting and computing the details of the map. It is 
desirable in conducting such surveys to establish a small field 
laboratory at a central point to which the camera and plates 
may be taken for the purpose of development, changes of 
plates, etc. (Chap. XLL) In making field surveys an out¬ 
line sketch of the terrane should be made in a note-book, on 
which memoranda must be made of names, roads, paths, 
buildings, and other information essential to the map. 

127. Projecting the Photographic Map. —Two drawing- 
boards are covered with paper, one of which is used as a con¬ 
structing board, on which the graphical determination of the 
points is made, and the other is used for the final drawing of 
the topographic map. On both are projected the trigono¬ 
metric points which are platted by means of their coordi¬ 
nates. The camera stations are platted on the board either 
by coordinates or by means of the protractor. The inter¬ 
mediate points are then projected by searching for well-defined 
points coming on two or more negatives, selecting such as 
seem most useful as guides for the drawing of the contours, 
and tracing the trend of mountain ranges, streams, etc. 


PROJECTING THE PHOTOGRAPHIC MAP . 


301 


Assuming that ten views have been taken panoramically 
from one station, then the horizontal projection of the ten 
plates exposed from such a station forms a decagon (Fig. 92), 
with a radius of inscribed circle equal to the principal focal 
length of the camera. After the position of one of these 
panoramic views has been found on the map by platting the 
angle from the occupied station to some located point, the 
orientation of the other point is accomplished by adding one 
tenth of 360° to this angle, and thus the entire decagon can 
be platted with reference to the occupied station and the 
orienting triangulation point. When this orientation of the 
horizontal plan is accomplished, the direction lines are drawn 
from the platted camera station to points photographed in the 
camera. The following example taken from the U. S. Coast 


4s 



Fig. 92.—Projection of Camera-plates from a Station. 

Survey report for 1893, by A. J. Flemer, further describes 
the process: 

Let mm'nn' (Fig. 93) represent a vertical and oriented 
perspective view, and 00 ' be the line of the horizon of the 
plate, V the point of view, and 00 the angle of orientation of 
the plate in reference to a secondary point A . Now, VP — f 
is the principal focal length, and if a is the representation on 
the plate of a point A in nature, and a vertical aa has been 





302 


PHO TOG PA TH/C S UR VE YING. 


drawn on the plate through it to the horizon line, then Pa' 
will be the abscissa of the point a . From the rectangular 
triangle VPa we have then 

x — f tan go .(24) 

In order to draw the horizontal position of the ray from V 
to A, the distance p' a' y equal to ;r, is laid off upon the hori¬ 
zon line 00 ' from P'. This distance a: is taken from the 
picture by means of a pair of dividers. The position of the 
point A' will be in the intersection of two or more lines of 
direction obtained in a similar manner from other pictures 
containing a and taken from other stations, and the same 
applies to all other points of the terrane if they can be 
identified upon plates taken from different panoramic stations. 


A 



Fig. 93. —Projection of Photograph. 


The elevations of points on the terrane are determined, 
after the selected points have been platted in horizontal plan 
as above, in the following manner: 

If the elevation of the camera station V is known, the 
elevation of the horizon line on the plate, inriy can be obtained 
by adding the height of the instrument to the elevation of 










COMPUTING POSITIONS . 



Fig. 94. Construction ok Map from Four Photographic Stations. 




































304 


PHO TO GRA PH/C S UR VE YING. 


the station V. The elevations of all points on the plates which 
are bisected by the horizontal line have the same elevation as 
the horizontal axis of the instrument at the station, dis¬ 
regarding curvature and refraction. The elevations of other 
secondary points selected from the plates are obtained by 
determining their elevations above or below the horizon line. 
From the relations of similar triangles we have 



in which h is the difference of elevation between the occupied 
station and the point observed, D the horizontal distance to 
the observed station A from the occupied station V, and d 
the horizontal distance of the same on the picture, y being 
the ordinate of points. From the rectangular triangle VPa' 
we find d — f sec go, when 



Dy 

f sec go 



The differences of elevation taken from the perspective are 
positive or negative according to the relative positions of 
their points in respect to the horizon. 

The computations and office platting connected with 
photo-topographic surveying are long and tedious operations, 
one day’s work in the field frequently requiring from four to 
eight days’ office work for the accomplishment of the platting 
of the map. In Fig. 94 is shown the mode of constructing 
of a portion of the map of the Canadian Survey. This was 
made from four camera stations, the view from one of which 
is shown in Fig. 90. 





PART III. 


HYPSOMETRY, OR DETERMINATION OF HEIGHTS. 


CHAPTER XV. 

SPIRIT-LEVELING. 

128. Hypsometry.—Hypsometry is that branch of sur¬ 
veying which treats of the determination of absolute heights 
or relative elevations. Mean sea level is the usual plane of 
reference from which such heights are determined, though 
not infrequently other arbitrary base levels are assumed for 
special purposes. There are three principal hypsometric 
methods, noted here in their order of accuracy, viz.: 

1. By spirit-level; 

2. By trigonometric or angular measurement; and 

3. By barometer or atmospheric pressure. 

Barometric leveling may be performed whenever the 

station the height of which is to be determined can be 
occupied; trigonometric leveling can be prosecuted when one 
or both of the stations is inaccessible; and spirit-leveling, only 
when both stations are accessible and visible one from the 
other. 

Hypsometry , or leveling , is the determination of the rela¬ 
tive elevations or heights above sea level of points upon the 
earth’s surface, and may be further classed as direct and 
indirect. Direct leveling is performed by the spirit-level and 
consists of the prolongation of a level line and the determina- 

305 



306 


SPIRIT-LE VELING. 


tion by actual measurement, on a vertical rod, of heights 
above or below this line. Indirect leveling is the determina¬ 
tion of heights by calculation from measured angles and 
distances or by barometric methods. 

Two points are said to be upon the same level when they 
are equidistant from the earth’s center. A level line is at a 
uniform distance from the equal potential surface, and, owing 
to the figure of the earth, the difference between the polar and 
equatorial levels is 13 miles vertical. A level line is not a 
horizontal line , for the latter is a straight line parallel to a 
tangent of the earth’s circumference, whereas a level line is a 
curved line, because it is parallel to the curvature of the sea. 
But for all ordinary purposes a level line and a horizontal 
line are synonymous even for leveling operations conducted 
over such great distances as to be affected by the curvature 
of the earth. A level surface may also be defined as one 
which is everywhere perpendicular to the direction of gravity 
as indicated by a plumb-line; and the spirit-level, like the 
plummet, is a device for utilizing the law of gravity to estab 
lish a horizontal or perpendicular line. 

129. Spirit-leveling.—The operation of spirit-leveling is 
the most accurate of hypsometric methods, because it is the 
simplest and most direct and is subject to the fewest sources 
of error in measurement or instrument. It is not dependent 
upon the exact measurement of horizontal distances nor of 
angles, nor is it affected by atmospheric changes. It is 
practically subject only to errors of instrument and level- 
bubble and of the staff or rod by which the vertical heights 
are measured. 

Spirit-leveling may for convenience be divided into three 
general classes: 

1. Ordinary or engineering spirit-leveling; 

2. Precise spirit-leveling; and 

3. Trigonometric or, as it is sometimes called, geodesic 
spiritdeveling. 


CLASSES OF SPIRIT-LEVELING . 


307 


The first two of these methods of spirit-leveling are essen¬ 
tially similar, differing chiefly in the care taken in the 
conduct of the work, and in the elimination or correction of 
instrument errors. In engineering spirit-leveling it is assumed 
that the adjustments of the instrument eliminate instrument 
errors, and no attempt is therefore made to correct these: 
moreover, the work is conducted with but moderate care, 
both in the quality of the instrument and rods employed, the 
turning-points upon which these rest, and in the various 
other phases of the operation of leveling. 

Precise spirit-leveling is conducted with finer instruments 
and rods and with all the care which it is possible to exercise 
in every detail of the work, especially in the elimination of 
the errors of instrument in the process of leveling. Account 
may or may not be taken of instrumental errors, and correc¬ 
tion may or may not be made for them, though with proper 
precautions to eliminate these more accurate results can be 
obtained than by attempting their correction, since the 
method of determining and compensating for such corrections 
involves other operations which may introduce counterbalanc¬ 
ing errors. 

Geodesic spirit-leveling accepts the instruments as in¬ 
accurate, and corrections are made for the instrumental 
inaccuracies by determining the instrument constants and 
applying them. Moreover, the operation is a combination 
of direct and indirect leveling, because, in addition to pro¬ 
longing the horizontal line as determined from the level- 
bubble, a slight angular measurement, calling for a correction 
to height dependent upon the distance, is introduced in each 
sight. This is done by making the instrument approximately 
level and reading the rods, then by making it truly level by 
a milled-head micrometer leveling-screw; the angular dis¬ 
tance through which the telescope is moved in the perform¬ 
ance of this operation, as recorded on the micrometer, is 
multiplied into the distance between the instrument and rod, 


3°8 


SPIKIT-LE VELING. 


and the resulting difference in height is a correction to the 
height directly measured by the instrument used as a spirit- 
level. 

130. Engineering Spirit-levels. — There are several 
methods of leveling according to the sequence of rod and 
instrument. In ordinary spirit-leveling the practice is to use 
one rod, to read a backsight upon it, then have the rod 
moved forward and observe a foresight upon it. The same 
methods may be employed, but with greater rapidity, by 
having two rodmen, so that immediately after the backsight 
is read on the rear rod this may be moved sufficiently in 
advance for the second foresight while the instrumentman is 
reading a foresight on the front rod (Art. 144). In addition 
to increasing the speed, this method gives a slight increase 
in the accuracy because of the rapidity with which the back¬ 
sight and foresight can be read, thus avoiding a possible 
settlement in the instrument between the two, but for ordi¬ 
nary purposes this method is too expensive. In addition to 
these two single leveling methods, duplicate leveling may 
be done with one rod or with two rods and one instrument 
(Arts. 143 and 144), and these methods are those commonly 
employed in precise leveling. 

A description of the engineer s spirit-level (Fig. 95) is 
superfluous in a work of this sort. The best results are pro¬ 
cured by using any good make of instrument of 18 to 20 
inches length. It must have a stout tripod, good glasses, 
and bubble graduated preferably to 10 seconds of arc. The 
ordinary bubble graduated to 20 seconds (Art. 147) increases 
the speed but slightly and is not nearly so accurate. 

131. Adjustments of the Level. — Before any of the 
adjustments of the level can be properly undertaken, the 
cross-wires must be focused by pointing them on an object 
and moving the diaphragm until a strong definition of them 
is obtained. The ordinary adjustments of the Y level are: 

1. The adjustment of the line of collimation, by which 


ADJUSTMENTS OF THE LEVEL. 


309 


the cross-hairs are brought into optical axis, so that their 
point of intersection remains on a fixed point during an entire 
revolution of the telescope on its wyes; 



2. The level-bubble must be brought parallel to the bear¬ 
ings of the wyes, that is, to the longitudinal axis of the 
telescope; and 





































































































3io 


SPIRIT-1E VELING. 


3. The wyes must be adjusted, that is, the bubble 
brought into position at right angles with the vertical axis of 
the instrument. 

To adjust the line of collimation, make the vertical cross¬ 
hair tangent to any vertical profile, as a wall, and then turn 
the telescope half-way round in its wyes. If the vertical 
cross-hair is still tangent to the edge selected, it is collimated. 
Select some horizontal line, and cause the horizontal cross¬ 
hair to be brought tangent to it. Again rotate the telescope 
half-way round in its wyes, and if the horizontal cross-hair is 
still tangent to the edge selected, it is collimated. 

Having adjusted the two wires separately in this manner, 
select some well-defined point which the cross-hairs are made 
to bisect. Now rotate the telescope half-way round in its. 
wyes. If the point is still bisected, the telescope is colli¬ 
mated. A very excellent mark to use is the intersection of 
the cross-hairs of a transit instrument. 

To adjust the level-bubble , bring the level-bar over two of 
the leveling-screws, focus the telescope upon some object 
about 300 feet distant, and put on the sunshade. Clamp the 
spindle, throw open the two arms which hold the telescope 
down in its wyes, and carefully level the instrument over the 
two level-screws parallel to the telescope. Lift the telescope 
out of its wyes, turn it end for end, and carefully replace it. 
If the level-tube is adjusted, the level will indicate the same 
reading as before. If it does not, correct half the deviation 
by the two leveling-screws and the remainder by moving the 
level-tube vertically by means of the two cylinder-nuts which 
secure the level-tube to the telescope-tube at its eye-piece 
end. Loosen the upper nut with an adjusting-pin, and 
then raise or lower the lower nut as the case requires, and 
finally clamp that end of the level-tube by bringing home the 
upper nut. Repeat until the adjustment is perfect. 

To make the level-tube parallel to the axis of the telescope, 
rotate the telescope about 20° in its wyes, and note whether 


TARGET LEVELING-RODS. 


311 

the level-bubble has the same reading as when the bubble 
was under the telescope. If it has, this adjustment is made. 
If it has not the same reading, move the end of the level- 
tube nearest the object-glass in a horizontal direction, when 
the telescope is in its proper position, by means of the two 
small capstan-headed screws which secure that end of the 
level to the telescope-tube. 

To make the level-bar parallel to the axis of the level- 
tube y level the instrument carefully over two of its leveling- 
screws, the other two being set as nearly level as may be; 
turn the instrument 180° in azimuth, and if the level indi¬ 
cates the same inclination, the level-bar is adjusted. If the 
level-bubble indicates a change of inclination of the telescope 
in turning 180°, correct half the amount of the change by 
the two level-screws, and the remainder by the two capstan¬ 
headed nuts at the end of the level-bar which is to the engi¬ 
neer’s left hand when he can read the maker’s name. Turn 
both nuts in the same direction an equal part of a revolution, 
starting that nut first which is in the direction of the desired 
movement of the level-bar. 

132. Target Leveling-rods.—Leveling-rods are of two 

general types: 

1. Target-rods; and 

2. Speaking or self-reading rods. 

These, again, may be extensible or of one piece. The 
three more usual types of target-rods are made in two pieces, 
one of which slides on the other so as to extend their length 
when in use, yet when not in use the length is reduced to 
one-half its possible limit for convenience in transportation. 
These three forms of rods are known respectively as the New 
York, Philadelphia, and Boston rods. Each of the two 
pieces of which these rods are constructed is about 7 feet in 
length, and the graduations are so arranged that the total 
extension possible with them is 12 feet. 

The New York rod (Fig. 96, a) is the best constructed and 


312 


SPIRIT-LE VELING. 


the most accurate of the three and is divided to hundredths of a 
foot, reading with the vernier on the target to thousandths. 



a b c 

Fig. 96.—Target-rods. 

The divisions are so arranged, however, that only those below 
-6J feet, that is, only those visible when the rod is not ex¬ 
tended, can be read from the instrument. On extension the 

























































































































SPEA KING-ROD S. 


313 


rod is read by a vernier on the rear side. The Philadelphia 
rod (Fig. 96, b) is divided to hundredths and is so graduated 
as to be easily read by the levelman at all distances at which 
it is visible. There is no vernier on the target of the Phila¬ 
delphia rod, so that the least reading practicable with it is 
one-half a hundredth, and by estimation perhaps to two- 
thousandths, of a foot. 

Unlike the rods just described, the Boston rod (Fig. 96, c ) 
has a fixed target, and all readings upon it are obtained by 
extending the rod. It is held with the target down for read¬ 
ings less than 5-J feet, and is inverted for greater readings. 
The vernier and the scales by which the rod is read are on 
the sides, and the divisions are such as to permit of its 
being read to one-thousandth of a foot. This rod is lighter 
and more compact than the others, but is not so' com¬ 
monly used. 

For very accurate work with a Nczv York rod, the foot¬ 
plate, instead of being the full width of the rod and of brass, 
should be a small truncated pyramid of phosphor-bronze or 
steel, the least dimensions of which at the bottom should be 
about one-half inch, in order that the rod when rested on the 
turning-point shall surely be balanced over its center and 
that the same point of the foot-plate should always be in 
contact with the turning-point. Great care should be taken 
to keep this foot-plate wiped clean, and in making extensions 
of the rod care should be taken that the vernier of the target 
is exactly set on the 6.5-foot mark when clamped. Also, 
after extension, care should be taken that no grit or dirt gets 
into either of the abutting joints, else readings taken between 
6 feet and 6.5 feet might be in error. Plumbing-levels 
should also be used where careful work is attempted. 

133. Speaking-rods.—The greater part of the leveling 
ordinarily done is of the more hasty and rougher kind, read¬ 
ings being taken on intermediate stakes to one-tenth foot only, 
and on turning points rarely nearer than one one-hundredth 


SPIRIT-LE VELING. 


3H 





foot. For this reason most levelmen prefer to use speaking- 
rods, and, as a consequence, of the extensible rod the 
Philadelphia is the more commonly used because it is also a 
speaking-rod. 

The non-extensible speaking - rods are, however, more 
easily and safely employed than extensible rods. They are 
more popular with the more experienced levelmen, as with 
them better work can be performed than with extensible 
speaking-rods. They are, moreover, frequently used in pre¬ 
cise leveling, as preferable to target and vernier rods. There 
are many modes of graduating speaking-rods so as to make 

the divisions legible at the greatest 
distance at which the rod is sighted. 
F'ew such rods can be purchased 
of instrument-makers, the easiest 
way to obtain them being for the 
levelman to divide and paint them 
himself. They consist usually of 
well-seasoned pine to 1 inch in 
thickness and from 3 to 5 inches 
in width. The fi gures are made as 
large as possible, so as to be legible, 
and various markings are introduced 
between these or in the shapes of 
the figures themselves, so that the 
eye shall have a guide whereby to 
divide the spaces (Figs. 81 and 97). 
In order to get more accurate 
a work from a speaking-rod the level 

Fig. 97 .— Speaking Level- should have three horizontal eross- 

RODS. 

hairs in the diaphragm, and the 
levelman should tell the rodman where to place his finger or 
pencil, and the latter should record this as an approximate 
check on the reading. The levelman should then read each 
of the three cross-hairs and record its reading separately, so 






















TURNING-POIN TS. 


315 


that by taking a mean of these he has a greater check on the 
leading observed and gets a more accurate determination of 
the height than by reading one cross-hair only. 

134. Turning-points.—In rough leveling it is of little 
consequence what manner of turning-point be used. The 
turning-point may be on a pebble or other hard object on 
the ground, or on a short stake driven into the ground, or a 
hatchet laid on the ground. Where accurate work is attempted 
a better turning-point must be employed, as the top of a rail, 
the head of a hatchet the blade of which is driven firmly into 
the ground, or a spike-shaped hammer, ora large stone which 
is well embedded in the ground. 

For precise work, however, these forms of turning-point 
are not sufficiently stable. Along railroads the best pos¬ 
sible turning-point is on top of the rail at a point clearly 
marked. Elsewhere two general forms have been em¬ 
ployed, one consisting of a hemispherical disk of iron, about 
6 inches in diameter, with short spikes on the under side 
which are pressed into the ground by the foot. (Fig. 98, a.) 
This form is approximately that employed by the British 
Ordnance Survey, but it is not believed to be as satisfactory 
as a long steel peg well driven into the ground. (Fig. 98, b.) 
Such pegs should be f to 1 inch in diameter at top and 



b 

Fig. 98.—Turning-points. 


from 12 to 18 inches in length, according to the consistency 
of the soil into which they are set. These should be firmly 
driven into the ground with a heavy hammer, a sufficient 
number of blows being struck to assure that the last few blows 
cause it to subside but little, and that friction is sufficient to 
















3 i6 


SPIRIT-L £ VELING. 


prevent its further subsidence by the weight of the leveling- 
rod. The turning-point should be made of hardened steel, 
and the top rounded and kept so by frequent dressing at a 
smithy in order that there shall be but one point of contact, 
and that the highest. 

135. Bench-marks.—In the course of any line of levels, 
be it short or long, accurate or approximate, marks should be 
left, the heights of which are determined by the leveling-rod, 
and these should be of such permanent character as not to be 
liable to mutilation or injury either accidentally or maliciously. 
This is in order that any future leveling which may be done 
in the neighborhood may start from or connect with the pre¬ 
vious level line; and in order that the point of connection 
may be fully identified, such marks must be left and be fully 
described in the notes. 

These bench-marks, as they are called, should be left 
preferably not farther apart than one mile, but may be farther 
than this or nearer together according, 1, to the character of 
the work; 2, the opportunity for description; 3, the purpose 

for which it is done ; and 4, the chances 
of connection with other lines of levels. 
On a line of railway they should be at 
such distance from the right of way as 
to assure their not being destroyed dur¬ 
ing construction of the road. Along 
highways or across country they should 
be so placed that they can be easily 
identified by descriptions which state 
their relation to some well known 
object (Fig. 99), and they should not 
be placed upon rocks, etc., which are 
liable to disturbance either by repairs 
to the highway or by work in the adjacent fields. 

One of the more common forms of bench-mark is a nail 
driven into the root of a tree. The nail should not be driven 



Fig. 99. 

Illustrated Descrip¬ 
tion of Bench-mark. 







METHOD OF RUNNING LINES OF LEVELS. 317 

into the trunk above the ground because of the difficulty of 
placing the rod upon it. The nail placed in the root should 
be as near to the trunk as possible, in order to guard against 
its being accidentally struck, and a notch should be so cut in 
the root as to leave one part of it a little higher than any of 
the surrounding wood, and into the highest point of the 
notched root the nail should be driven flush to its surface. 
The best nail for such purposes is one of copper, as it can 
always be surely identified as distinct from nails which may 
accidentally or maliciously be driven in its neighborhood. 
Next to copper nails, wire nails are most satisfactory as 
bench-marks. 

The corner-stone or water-table of a building, a door-sill, 
abutment of abridge, or massive rock pier, all furnish desirable 
sites for bench-marks. The exact spot should be marked by 
a chisel-cut. For more permanent bench-marks, such as are 
left in precise leveling, it is customary to drill a hole in solid 
rock or the foundation-stone of some stable structure and to 
place a copper bolt in this. For greatest security from subsi¬ 
dence a building had better not be used, but a stone or iron 
post should be planted deep into the earth and the top of this 
be used as a bench-mark. 

Fig. ioo shows a form of iron post used by the U. S. 
Geological Survey. Under this, in the bottom of the hole, is 
placed a large flat stone. This post is cheap and light, as it 
is of wrought-iron pipe. The same organization uses bronze 
tablets of similar design for cementing in masonry walls. In 
Germany small wrought-iron pins with round heads are ce¬ 
mented into walls or posts for bench-marks. 

136. Method of Running Single Lines of Levels—In 
general the details of the most approved methods of running 
spirit levels, as practiced by the more successful levelmen, 
may be stated as follows: 

The rodman , after examining and wiping the bottom of 
the level-rod, standing behind it, balances it vertically on a 


3 i8 


SPIN IT-LE VELING. 


bench-mark or a steel turning-point firmly driven into the 
ground. He waves it back and forth gently as he balances 
it, so that the levelman may see (hat it is plumb in the direc¬ 
tion of the line of sight, and the latter calls to him, not by 
signaling with the hand, but by word of mouth, the exact 



Fig. ioo.—Bronze Tablet and Wrought-iron Bench-mark Post. 

figures on which to set the target. The rodman then takes 
down the rod, sets the target, clamps it and again holds it on 
the turning-point, when the levelmen may call to him to raise 
or lower it one or more thousandths. Reclamping the target 
as directed, he now levels the rod carefully by watching the 
fore and back plumbing-level, the levelman waving him to 
level it across the line of sight as indicated by the vertical 
cross-hair. 

The levelman, having his instrument well planted, and 
sighting first at the rod and then examining the level-bubble, 
if he finds the target exactly set at the same time that the in¬ 
strument, as shown by the bubble, is exactly level, calls out 







































METHOD OF RUNNING LINES OF LEVELS. 3 19 

<l plumb,” which expression, or some equivalent thereto, is 
instantly repeated by the rodman if he finds his rod plumb, 
and if the target is then perfectly set, the levelman gives the 
signal “all right if not, he calls again to the rodman the 
amount by which the target is to be raised or lowered, and 
the same operation is repeated until the rod is found to 
be precisely plumb at the same instant that the instrument is 
level and the horizontal cross-hair bisects the clamped target. 

The rodman reads the rod and records his reading before 
he removes his turning-point, then he shows the rod to 
the levelman as they pass, the latter reading and recording 
the same. Both at once compute the height of instrument 
and compare results without having made any remarks one 
to the other concerning the rod reading, and if the results 
differ, as stated in the instructions, they then both reread 
the rod, recompute, and if the difference still exists, they 
must go back to the nearest bench-mark and rerun that 
much of the line. When a target setting falls above the 
6.5-ft. mark, at which point is the break in the jointed rod, 
the closure of the rod at this point must be examined by both 
to make sure that it is perfect, otherwise the joint must be 
cleaned or a correction made for the failure of the rod to 
properly close. 

In running a long single-rodded line of levels, the following 
additional precaution may be taken. Instead of setting and 
reading the target once, it is set and read twice; that is, it is 
double-targeted by the levelman first signaling the target to a 
setting up or down, when it is clamped, read and recorded by 
the rodman, who then loosens the target, continues to move it in 
the same direction in which it was going for, say, a tenth of a 
foot, when he is then signaled to a setting in the opposite 
direction. This gives a double target reading on each turn¬ 
ing-point, and the method of making these tends to eliminate 
parallax in target-clamping. If the two readings differ by 
more than two thousandths of a foot, additional settings are 


320 


SPIRIT-LE VELING. 


made. As the rodman and the levelman pass, the latter 
reads the target, which is left clamped at the last setting, and 
the rodman, though he records all readings, uses in his com¬ 
putations only the first of the pair adopted, while the level- 
man uses the last. 

137. Instructions for Leveling - .—The following are the 
instructions for levelmen issued by the Director of the U. S. 
Geological Survey: 

1. Primary level lines should be run with one or two rodmen and one 
levelmen, and when necessary a bubble-tender. Where such lines are run 
in circuits which will check back upon themselves or other lines, one rod- 
man will suffice. Where long, unchecked lines are run, two rodmen must 
be employed. 

2. Single-rodded Lines. —Levelman and rodman must keep separate 
notes and compute differences of elevation immediately. As levelman 
and rodman pass, the former must read the rod himself, record and com¬ 
pare readings, then compute the H. I., and after computations are made 
compare results with the rodman. No comparisons should be made until 
the record is complete. If the results differ, each must read the rod be¬ 
fore comparing anything but results. 

3. Work on primary lines should not be carried on during high winds 
or when the air is “boiling” badly. During very hot weather an effort 
should be made to get to work early and remain out late, rather than to 
work during midday. 

4. Foresights and backsights should be of equal length, and no sight 
over 300 feet should be taken excepting under unavoidable circumstances, 
as in crossing rivers at fords or ferries or in crossing ravines. In such 
cases extraordinary precautions must be taken, as repeated readings at 
changed positions of rod and level, etc. 

5. If it is impracticable to take equal foresights and backsights, as soon 
as the steep slone is passed take enough unequal sights to make each set 
balance In this case extra care must be taken to insure correct adjust¬ 
ment of the level. 

6. Distances along a railroad can be obtained by counting rails; at 
other times stadia or pacing may be used, according to the quality of the 
work. The distances in feet of both the foresights and backsights must 
be recorded in both note-books in the proper columns. 

7. The tripod clamping-screws must be loosened when the instrument 
is set, and tightened again only after the legs are firmly placed. Always 
level the instrument exactly before setting the target. After setting it 
and before giving the signal “ all right,” examine the level-bubble. If 
found to be away from center, correct it and reset target. 


INSTRUCTIONS FOR LEVELING. 


321 


8. The level must be adjusted daily, or oftener if necessary. The 
adjustment of the line of collimation and of the level-tube is especially 
important. 

9. Provide rodmen with conical steel pegs, 6 to 12 inches long, with 
round heads, to be used as turning-points. Never take turning-points 
between ties. Always drive the pegs firmly into the ground. In running 
along railroads the best results will be obtained by marking a spot on 
top of rail and setting the rod on it as a turning-point. 

10. When the rod is lengthened beyond 6.5 feet, both the rodman and 
the levelman must examine the setting of the target as well as the reading 
of the rod vernier. When the rod is closed see that the rod vernier indi¬ 
cates 6.5 feet, not depending upon the abutting end to bring it back to 
place. Keep the lower end of the rod and the top of the turning-point 
free from mud and dirt. 

11. Plumbing-levels must always be used and kept in adjustment, and 
long extensions of the rod avoided. 

12. Leave temporary bench-marks at frequent intervals, marked so 
that they can be easily identified. These may be on a solid rock well 
marked, a nail driven in the root of a tree or post, or on any place where 
the mark will not be disturbed for a few weeks. One such bench-mark 
should be left for every mile run, in order to give sufficient points to which 
to tie future levels. Mark in large figures, in a conspicuous place when 
possible, the elevation to the nearest foot. Make notes opposite all ele¬ 
vations at crossings of roads, railroads, streams, bridges, and in front of 
railway stations and public buildings, and of such other facts as may aid 
the topographer in his work. 

13. All permanent bench-marks must be on copper bolts or bronze 
tablets let in drill-holes in masonry structures or in solid rock, or be on 
the iron posts adopted by this Survey. The figures of elevation must 
be stamped well into the metal, to the nearest foot only, also name or 
initial letter of the central datum point. 

14. A complete description, accompanied by a large-scale sketch, 
must be made of each bench-mark, giving its exact elevation as computed 
from the mean of the two sets of notes. After bench-marks are stamped 
both levelman and rodman must examine them, and record in note-books 
the figures stamped thereon. 

1 5. The limit of error in feet should not exceed .05 V distance in miles. 

16. Use the regular Survey level-books; keep full descriptive notes 
on title-page of every book, giving names, dates, etc. Each man should 
be responsible for his own note-book ; and under no circumstances should 
erasures be made, a single pencil-line being d-rawn through erroneous 

records. 

17. When errors are discovered as the work progresses, report the 
same at once to the topographer in charge. 



322 


SPIRIT-LE VELING. 


18. Keep each set of notes separately and independently as taken, 
paying no attention whatever to other notes except to compare results. 
If on comparison errors are discovered, correct them only by new obser¬ 
vations or computations. All notes must be recorded directly in note¬ 
book. Separate pieces of paper for figuring or temporary records must 
not, under any circumstances, be used. 

19. In long, single-rodded lines make two target-settings on each 
turning-point, by first signaling “ up ” or “down ” to a setting, which is 
recorded by the rodman, then unclamping and signaling in the opposite 
direction to a setting. If the two differ more than .002 of a foot, addi¬ 
tional readings must be made. The rodman should record all readings, 
using in his computations only the first of the pair adopted, and the 
levelman the last. 

20. Double-rodded Lines. —In running unchecked or single primary 
lines with two rodmen, they should set on turning-points ioto 20'feet apart, 
but each at equal distances for foresights and backsights; otherwise the 
above instructions are to be followed with the following modifications : 

21. The tripod clamping-screws should be loosened when the instru¬ 
ment is set, and tightened only after the legs are firmly planted, and the 
instrument must be shaded at all times by the bubble-tender. 

22. The laborer should place the steel turning-points for foresights 
and then return and not remove the backsight points until the levelman 
has set targets on the new foresight, so that there shall be in the ground 
at all times two turning-points the elevations of which are known. 

23. Bench-marks left at termination of work at night, or for rain or 
other caus;% should be practically turning-points in a continuous line. 
They should consist of large wooden pegs driven below the surface of the 
ground, with a copper nail firmly embedded in the top. One of these 
pegs is to be used as the final turning-point for each rodman. They are 
to be covered with dirt or otherwise hidden, their location being marked 
by sketches in note-books showing relation to railroad ties, telegraph- 
poles, etc. 

24. An index-book or list of bench-marks must be kept posted in the 
field, in ink, for all classes of leveling done. In these, location sketches 
of permanent bench-marks may be made, and descriptions should in 
every case refer, with distance, to some village, section corner, or other 
place of local importance. All circuit-closure errors should be distinctly 
noted, with cross-reference by page to the connecting lines. 

138. Note-books. —Where the leveling is for a line of rail¬ 
way or canal, elevations are taken at every one hundred feet 
and at intermediate points to note sudden changes in slope or 
at stream crossings and similar features. The more usual ruling 


NO TE-BOOKS. 


323 


in a level note-book is to have one page divided into several 
columns, the opposite page being left free for remarks and for 
a plot of the level line, showing position of turning-points,, 
road a^d stream crossings, etc. 

LEVEL NOTES. 






Date, Sept. 26, 

1898. 

Dist. B. S. 

Dist. F. S. 

Backsight. 

H. I., Feet. 

Foresight. 

Elevation, 

Feet. 

Sta. 



Morehouse 

ville to Pis 

eco, N. Y. 







7-933 

I910.429 

I 

55 

55 

10.721 

1921.150 

1913.212- 


20 

45 . 

0.786 

1913.998 

11.984 

I902.014- 


23 

55 

0.801 

1902.815 

IO.587 

1S92.228 

2 


In the first or station column are placed the letters “ B. M.” 
with number, for bench-marks, and “ T. P.” with station num¬ 
ber, to indicate the position of turning-points. In the backsight 
column is placed the reading observed in backsighting on any 
bench-mark or turning-point. In the height of instrument col¬ 
umn is placed the height of the line of collimation of the instru¬ 
ment as obtained by adding to the last recorded elevation in the 
fifth column the reading of the rod recorded in the backsight 
column. In the foresight column is placed the reading of 
the rod recorded at each of the intermediate stations, and 
next to it in the elevation column the elevation is obtained 
by subtracting the foresight from the height of instrument; 
also the reading of the rod at the foresight on the next turn- 
ing.point or bench-mark is obtained by subtracting the fore¬ 
sight from the height of instrument. Not uncommonly the 
notes in the book are kept by having the foresight and eleva¬ 
tion of the next turning-point recorded on the line below that 
on which the backsight and height of instrument and the last 
turning-point are recorded. 

139. Platting Profiles.—For purposes of construction and 
in order that levels may be more readily interpreted, the 
notes are platted on what is called cross-section or profile 


















324 


SPJRIT-LE VELING. 


paper so as to show graphically the undulations of the surface 
passed over. There are numerous forms of ruling for profile 
papers which are kept in stock by various dealers in mathe¬ 
matical instruments, the more common being a vertical ruling 
which divides the paper horizontally into spaces about J inch 
apart, while the horizontal divisions or elevations are shone 
by vertical spaces of like size, but heavily ruled and divided 
into five smaller spaces by finer ruled horizontal lines. 

In platting the profile a convenient elevation is assumed 
for the bottom horizontal line, perhaps sea-level or some 
datum which will be the lowest point on the line of the route 
leveled, and opposite it maybe marked zero as datum or its 
elevation above sea-level, if this is known. For railway or 
canal work where construction is to follow, it is usual to 
assume one foot as the smallest vertical interval of the profile- 
paper, and io feet as the smallest horizontal interval, the 
proportion then being 5 feet of vertical to 1 of horizontal, or 
5 to 1. Various other proportions may be used, a greater 
disproportion of vertical to horizontal being employed to ac¬ 
centuate the irregularities of very rough country, each hori¬ 
zontal division being assumed as 10 feet, or 100 feet, or a 
fraction of a mile, as the case may be. The distance to each 
turning point or station at which the elevation is determined 
is that ascertained by counting the vertical lines from left to 
right, and above it the corresponding elevation is platted by 
counting from the datum or base line the proper number of 
horizontal lines. 


CHAPTER XVI. 


LEVELING OF PRECISION. 

140. Precise Leveling - .—When for any reason it is neces¬ 
sary to determine elevations with the greatest precision attainable, 
as in government work along the Mississippi and Missouri rivers, 
and where elevations have to be carried great distances from 
the ocean, in order to give datums on which to base other levels 
as the primary level of the U. S. Geological Survey or the geodetic 
investigations of the L T . S. Coast Survey, spirit-leveling is executed 
by methods which differ materially from those just described. 

In the United States three methods of precise leveling have 
been practiced by three different government organizations. 
One of the oldest and most satisfactory is that employed by 
the U. S. Engineers on the Mississippi River, and is an adapta¬ 
tion of the European modes of leveling, in which a Swiss instru¬ 
ment, the Kern level, is used and a speaking-rod is employed. 
The U. S. Coast and Geodetic Survey devised a peculiar instrument, 
called a “ geodesic” level, which was exclusively used by them 
in connection with the target-rod prior to 1900. The U. S. 
Geological Survey used, prior to 1904, a modification of an in¬ 
strument originally designed by Mr. Van Orden of the Coast 
Survey, and employed purely spirit-leveling methods, usirg 
either target- or speaking-rods. In 1900 the Coast Survey de¬ 
signed and adopted a binocular prism level with speaking-rods 
graduated to meters, which has given more successful results 
than ever before attained. In 1904 the Geological Survey 
adopted the same instrument but with a speaking-rod divided 
to yards, for convenience in reducing to feet by dividing by three 
for the three horizontal wires. The discarded geodesic level 

325 


326 


LEVELING OF PRECISION . 


is fully described in various reports of the Coast and Geo¬ 
detic Survey, as well as in Johnson’s “Surveying” and Baker’s 
‘‘Engineer’s Surveying Instruments,” and will, therefore, not 
be described in detail here. 

141. Binocular Precise Level.—In 1900 a new type of 
precise level was adopted by the United States Coast and Geodetic 
Survey, which is doubtless the most accurate instrument ever 
devised for the work. The principal characteristics which 
distinguish it are the irreversibility of the telescope and level, 
the absence of Y’s, the rigid fastening of the level-vial to the 

telescope and its close juxtaposition 
to the latter, in the barrel of which 
it is countersunk; the use, in the 
construction of the telescope and 
adjacent pirts, of a nickel-iron alloy 
having a very small coefficient of 
expansion; the protection of the 
level-vial and the middle part of 
the telescope from sudden and 

Fig. i 00a.— Section of Binocular unequal changes of temperature by 

Precise Level. ,1 , , . . 

incasing them m an outer tube; and 

an arrangement by which, without any change of the observer’s 

position, the level-bubble can be clearly seen by his left eye at 

nearly the same instant in which the distant rod is observed 

through the telescope by his right eye. 

It has been found that one of the principal sources of error in 
some of the precise leveling of the past was due to unequal changes 
of temperature in the telescope and parts connecting it with the 
level-vial. The momentary changes in the relative positions 
of the level-vial and telescope produced by these temperature 
changes, though microscopic in magnitude, were yet sufficient 
to introduce appreciable errors into the leveling on long lines. 
The particular nickel-iron alloy used in the construction of the 
new level expands less than one-fourth as much as brass for a 

















PRECISE SPIRIT-LEVEL. 


327 


given increase of temperature. This, together with the fact 
that the level is mounted as close as possible to the line of sight, 
gives a very high degree of stability to the relation between the 
two. 

142. Precise Spirit-level.-—The adjustments of precise levels 
do not differ essentially from those of ordinary Y levels. In 



Fig. 1006.—The Coast Survey Binocular Precise Level. 

the latter the less important adjustments are neglected as 
being less than the degree of accuracy aimed at; but as 
extreme accuracy is desired in precise leveling*, every adjust¬ 
ment must be carefully made, even though the instrument is 
used in such manner as to eliminate errors of adjustment. 
Accordingly, the instrument is adjusted as nearly as prac- 






328 


LEVELING QF PRECISION. 


ticable, and then the errors of instrument are determined and 
each single observation corrected for these errors. As the in¬ 
equality of the diameters of the collars cannot be eliminated by 
a system of double observations, since the line of vertical axis 
is invariable, it is practically eliminated from the final result 
by reading equal foresights and backsights. Although the 
inequality of the diameters of the collars cannot be eliminated 
by double readings, it can be determined by observations with 
a striding-level, as in the case of the astronomic transit, and 
can be applied as a correction to the rod readings where a 
system of double-rodding is employed. 

The precise level used by the U. S. Geological Survey 
prior to 1904 is made by Messrs. Buff and Berger of Boston. 
One of the features of this level is a very firm tripod with split 



legs, so as to give a broad head and correspondingly firm base 
for the support of the instrument. (Fig. 101.) On the head 
the level is supported freely by three leveling screws, and it is 
clamped to the tripod-head by a stout center screw when not 




























































































INSTRUCTIONS , PRECISE LEVELING. 


3 29 


in use. The telescope has an aperture of i| inches and magni¬ 
fying power of 40 diameters, and is inverting. It likewise ro¬ 
tates in vertical plane by means of a milled-head screw nearly 
under the eyepiece. 

143. Instructions for Precise Leveling. —The following are 
the instructions governing precise and primary leveling in the 
U. S. Geological Survey, using prism level and yard rod: 

1. The stadia value and value of thread intervals should be determined 
by observation on a measured base before using level. The extreme wires 
should be fixed to cover 2 yards at 1000 feet. The mean of the two thread 
intervals in thousandths of a yard will then be the distance to rod in feet. 

2. The distance between successive standard bench-marks should not 
exceed 4 miles and must not average more than 3 miles. 

3. Standard bench-marks along a railroad or highway should be placed 
outside the right of way. 

4. Endeavor to set standard bench-marks within one-tenth foot of 
the marked elevation. The figures of elevation to the nearest foot only 
must be well stamped into the metal cap. 

5. Place standard bench-marks near all important lakes and reservoirs. 
Wherever it is impossible to make the standard bench-mark a turning point 
in the line, two temporary marks must be left and the line tied to both 
when setting the standard mark. Connect when possible with two or more 
bench-marks when starting or closing a line. 

6. Secondary bench-marks may be chiseled squares or crosses on solid 
rock or masonry, or a nail or spike may be driven in a telegraph pole, mile¬ 
post, or tree. Select a place where the mark will not likely be disturbed, 
preferably near road junctions. These should be conspicuously marked 
with large figures, with white or red paint. The proper marking is 

US 

(elevation). Where there are no natural objects for bench-marks, pieces 
B M 

of one-inch iron pipe, about 20 inches long, may be used. Make full notes 
opposite all elevations, at crossings of railroads, summits, bridges, and 
junction of roads, and in front of railway stations and public buildings. 
The date, time, and depth of water should be included in descriptions re¬ 
lating to water-surface elevations. 

7. In primary work, contour crossings should be indicated by marked 
stakes or otherwise. The same side of the road, preferably the north or 
east, should always be used for these markings. 

8. Establish temporary bench-marks at distances ranging from % to 
i£ miles. 

9. Connect lines with all possible bench-marks of cities, railroads, or 
other organization. 


330 


LEVELING OF PRECISION . 


io. Old bench-marks are to be fully described. For precise leveling 
computation, use old name or letter. 

n. Complete descriptions of bench-marks and useful elevations must 
be kept in ink in field note book 9-940 and should be copied in description 
book 9-916 at close of each day’s work. A sketch should be made of 
standard bench-mark locations in both books, and descriptions written in 
the following form: 

First. Name of the nearest post-office, town, or village, or other well- 
known named locality, with direction and distance to the bench-mark in 
miles and tenths; or, township, range, and section in which bench-mark 
stands, with direction and distance from nearest corner. 

Second. Position with reference to buildings, bridges, mile-posts, street 
or road corners. 

Third. Description of object on which the bench-mark is placed—tree, 
fcoulder, bridge, etc. 

Fourth. The nature of the bench-mark, whether bolt, mark on rock, 
aluminum tablet, post, etc., and how marked or stamped. For standard 
bench-marks, give figures and datum letters. 

12. Keep descriptions in the order in which the bench-marks occur; 
if standard bench-marks are not established when the line is first run, spaces 
should be reserved in description books for them in their proper order. 
Give at frequent intervals a brief description of line, especially when chang¬ 
ing direction. When circuits are closed give complete descriptions of 
closing point, closure, old and new elevations, and page reference to con¬ 
necting points. Make a plat of all lines or circuits on a page near the back 
of the description book for each atlas sheet or group of circuits and indicate 
the names of enough places to readily identify the line. If there are public 
land surveys show the position of the line with reference to the township 
and section lines. Alongside of each line give reference to page where 
notes are recorded. Levelmen should mark the elevations on a map, also 
the page and book in which the notes are recorded. This map is to be filed 
w T ith field material. 

13. All new bench-marks on precise-level work are to be designated by 
capital letters for computation reference, with subscripts if necessary. On 
primary work no such designation is required. 

14. The elevation of the top of rail in front of each railroad station 
junction, and important switch, and at secondary stations without build¬ 
ings opposite sign boards, must be determined by reading from the nearest 
instrument station, and recorded with “backward” or “forward” added 
to show direction to rod from instrument. 

15. Take turning points on marked points on top of rail when leveling 
along railroad. 

16. Provide rodmen with steel turning-point pin, to be secured on 
requisition, to be used away from railways. 

17. The maximum length of sight permissible is 360 feet, and this maxi- 


DOUBLE-RODDED LEVELING. 


33 1 


mum is to be attained only under the most favorable conditions. An 
exception to this rule may be made when crossing rivers or deep ravines; 
in such cases proceed thus: Establish turning points on both sides, set up 
the level about 20 feet from each turning point in turn, taking in first position 
a backsight to near, and foresight to distant point; then cross stream or 
valley and take a backsight to distant, and foresight to near point. The 
mean of these determinations of elevation will be the true one. The instru¬ 
ment should be mounted and stakes so placed that the center wire will 
fall about center of rod and if the greater sight exceeds 1500 feet improvised 
targets should be made of cardboard and rubber bands and several settings 
made by raising and lowering an equal number of times to image of center 
wire. 

18. The level must be shaded by an umbrella during observations and 
by a cloth hood when carried between stations. 

19. The recorder must compute and record the thread intervals for 
backsights and foresights, also the continuous sums, on precise leveling, for 
the section between bench-marks, and for the page on primary leveling. 

20. In primary work endeavor to equalize sums of foresight and back¬ 
sight intervals daily. 

21. In precise work the maximum allowable difference between a back¬ 
sight and the corresponding foresight mean-thread interval is 0.033 yard 
(=33 feet distance). The continuous sums of rod intervals for the section 
between bench-marks must not be allowed to differ more than 0.132 yard 
( = 66 feet distance), and they should be kept as nearly equal as practicable. 

22. Primary level lines must be in circuits when practicable. When 
not, each line must be rerun in opposite direction. Errors of closure of 

circuits in feet should not exceed o.oqVlength of circuit in miles, which 
equals 0.056^distance between bench-marks in miles, for forward and back¬ 
ward lines. 

23. Precise level lines must consist of lines run independently in both 
the forward and backward direction. The allowable error in feet is 

0.017 's/distance between bench-marks in miles, and when this limit is 
exceeded on any section the forward or backward measure is to be repeated 
until a pair run in opposite directions is obtained between which the di¬ 
vergence falls within that limit. The last set-up of one running must not be 
used as the first set-up of a return running. 

24. On precise work, if any measure over a section differs more than 
0.02 foot from the mean, that measure shall be rejected. No rejection 
shall be made on account of a residual smaller than 0.02 foot. 

25. On precise level work, whenever a blunder, such as a misreading of 
one yard or one-tenth, or an interchange of sights, is discovered and the 
necessary correction applied, such measure may be retained, provided there 
are at least two other measures over the same section which are not subject 
to any uncertainty. It is especially desirable to make the backward 





33 2 


LEVELING OF PRECISION 


measurement in an afternoon, if the forward measurement was made in a 
forenoon, and vice versa. The observer should secure as much difference 
of atmospheric conditions between the measurements as is possible without 
materailly delaying the work for that purpose. 

26. At alternate stations on precise work the foresight is to be taken 
before the backsight, i.e., always take readings upon the same rodman 
first. 

27. When the level is on the tripod be sure that the central clamp screw 
is tight. 

28. Keep the telescope off of the micrometer-screw bearing while carry¬ 
ing between stations. 

29. Leave the three tripod wing-nuts loose when carrying; clamp tight 
when tripod is in place for work. 

30. The program at each set-up is as follows: After the tripod is firmly 
set and the clamp-screws tigthened, level. approximately by the circular 
level (which has been adjusted by comparison with the long level). Point 
instrument towards rod and clamp; bring level bubble to center of tube 
by means of the micrometer screw. Read on rod, taking first the color 
initials for the lesser and greater extreme readings; second, yards and tenths 
for each wire, taking smallest readings first; third, repeating and reading 
yards, hundredths, and estimated thousandths. Before level is moved 
recorder should first notice that color agrees with yard readings; second, 
compute the two thread intervals and if their ratio differs more than one 
per cent from the true ratio, the levelman must repeat the readings; third, 
compute the mean reading in feet by summation and test units and tenths 
by mentally multiplying the middle reading by 3. After an agreement is 
reached proceed similarly in taking the other readings. 

31 When work is commenced, and at least once a day thereafter, the 
adjustment of the level must be tested by the “peg method” as follows: 
At some convenient set-up, after the usual backsight and foresight readings 
have been recorded, copy the foresight on a separate line as a new foresight, 
leave the foresight pin in place, and set a second turning pin about 30feet 
back of the instrument; read rod on it for a new backsight; find from these 
distant and near rod readings the mean readings in feet as usual. Move 
the level forward to a set-up about 30 feet back of foresight pin, take Read¬ 
ings on foresight and then on backsight pin. The constant C, which is a 
factor of the adjustment correction, must be determined thus: 

Sum of near-rod readings in ft.—Sum of distant-rod readings in ft.* 

3 (Sum of distant-rod intervals in yds.—Sum of near-rod intervals in yds.) 

The rod interval to be taken for any sight is the difference of extreme 
wire readings. 

When the sum of the near-rod readings is the greater, the sign of C will 
be + and vice versa. Great care must be taken in pointing off decimals 
and in giving proper signs. If the resulting value for C numerically exceeds 
0.005, an adjustment should be made by changing the position of the level 


* A correction must be applied to the sum of distant-rod readings for curvature and 
refraction. See table XXXI, p. 549 . 




COMPUTING PRECISE LEVELING. 


333 


EXAMPLE OF COMPUTATION OF C 
As actually to be made in the field in accordance with the instructions. 
Determination of C, 8.20 a.m., Aug. 28, 1905. 


Backsight. 

Foresight. 

Thr. Rdg. 

Thr. Int. 

Sum T. R. 

Sum T. R. 

Thr. Int. 

Thr. Rdg. 

I- 5 I 5 

I .528 

0.013 

0.014 



0.105 

0.104 

0-357 

0.462 

I .542 

0.027 

4-585 

1.385 

O.209 

0.566 

2.252 

2-357 

2.462 

0.105 

0.105 



0.012 

0.013 

I . 276 
1.288 

I .301 

0.210 

0.209 

7.071 

1.385 

3-865 

4.585 

O.025 

O.027 


O.419 

O.O52 

8.456 

-0.0005% 

8.450 

8-4555 

O.052 



0.367 yds. 

3 

4-4555 

1.101)- .oo55(-0.005 




1.101 feet 






q ’ 4 I 9 = 2io feet sum far distances, 
o 002 


bubble only, as follows: Point to a distant rod with the bubble in the middle 
of the tube and read; move the telescope (by micrometer screws) so as 
to raise the middle cross wdre by an amount which in yards is equal to C 
times the extreme wire interval. While holding the telescope in this 
position, bring the bubble to the middle of the tube by raising (or lowering) 
one end of the level vial with the adjustment wrench; if C is negative, the 
middle wire must of course be lowered on the rod. After the adjustment 
has been made, its accuracy should be tested by redetermining the value 
of C. It is desirable to have the determinations of C made under average 
conditions. 

32. In case the threads break, and the level-tube adjustment has not 
been disturbed, insert new spider threads and determine a value of C as 
above directed. Compare with last determination of C and adjust for the 
difference by changing position of the ring only—not the level bubble. 

33. On precise-level work, when commencing work for the day, and at 
beginning and end of each section, record time. Record a temperature 
for each set-up, using thermometer readings alternately for each rod. 

34 On primary work record time and temperature each hour. 

35. At the beginning and end of the season, and at least twice each 
month during the progress of the leveling, the 3 or 3.5 yard interval be¬ 
tween the metallic plugs on the face of each level rod must be measured 
carefully in feet to the nearest thousandth, always the same with tape kept 
for that purpose. Record also the temperature. 


































333 « 


LEVELING OF PRECISION. 


36 The rods must always be kept covered when not in use. Never 
let painted sides touch the ground. Should difficulty be found in holding 
a rod steady because of wind, two pieces of bamboo or other light poles, 
8 ft. long may be held by rodman against the rod so as to make triangular 
brace against the wind. 

37. Always keep the plumbing levels in adjustment. 

38. Computations required must be made at bottom of each page of 
primary notes, and at end of each section of precise work, each night or 
oftener, if convenient, by both levelman and recorder, independently. Sum 
only the center values of columns 1 and 7 and obtain a check by multiplying 
the sum of three and adding the algebraic sum of excesses of lower over upper 
intervals, which should equal the sums of columns 3 and 5 respectively. 

39. In precise leveling it is not necessary to complete the H. I. and 
elevation column, but the difference of elevation for each section should 
be computed. 

40. The field computations and abstracts (Forms 9-932 and 9-937) 
must be kept up as the work progresses. As soon as each book of original 
records is out of use it must be sent to the office by registered mail, and 
the corresponding abstracts retained in the field until information is re¬ 
ceived of the receipt of the original record. 

Special Instructions for Use of Prism-level Note Books No. 9-940, 
when Used for Primary and Precise Leveling Record. 

Fill in blanks on fly-leaf the first day the book is used. 

Fill in blanks at head of each page each day on precise work, denoting 
bench-marks run between by their letter or number, and the direction 
backward or forward with regard to direction of extending double line. 

Each horizontal space between two red lines is for a single set-up of 
the level. 

The notes for each section of line on precise work must be complete in 
themselves and commence on a new page. Every primary-line record 
must begin on a new page and initial bench-mark be fully described. 

Counting the columns from the left each is used as follows: 

Col. 1 is for the readings on rod in yards for the three stadia wires, the 
first recorded reading being for the wire giving lowest value. The color 
letter is to be placed alongside the first and last reading. The recorder 
should notice whether the color as recorded corresponds with the unit called 
out by the levelman, and he will be held responsible for errors between the 
recorded color initials and units. Each day the levelman should verify 
the comparison. 

Col. 2 is for the thread intervals for the thread readings in column 1, 
the upper ones being the difference between the lowest readings and the 
middle ones, the lower being the difference between the middle and the 
greatest readings of each set. (See also a following paragraph.) 

Col. 3 is for the sum of thread readings in column 1 between the two 
horizontal ruled red lines, and is equal to the mean in feet of the three 
readings on rod. 

Col. 4 with the exception of the last line is not intended for use under 
precise-leveling instructions, but can be used to compute approximate 


SEQUENCE IN LEVELING. 


3336 


elevations, filling out only at bench-marks. On primary work the first 
entry on the page at the left of the words “Elevation brought forward from 

page.should be the elevation from a previous page, or from another 

book. In the latter case, give book number and page, and in every case 
carefully verify the copying. The second entry, below the red line and 
above the short black line, is to be the height of the instrument as found 
by adding the first entry in column 3 to the first elevation in column 4. 
The third entry in column 4 is the elevation computed by subtracting the 
first foresight from first H. I In each case the H. I. will always be above 
a short black line and the elevation always just above a red line. 

Cols. 5, 6, and 7 are for foresight readings, corresponding with 3, 2, and 
1 for backsight readings. 

Col. 8 is for record of temperature and time. 

Col. 9 is for correction of curvature and refraction for unequal sights; 
need not be filled out in the field. 

Col. 10 is for extra foresights at points which are not turning points; 
also for their sum. 

Col. 11 is for description of bench-marks, for elevations from extra fore¬ 
sights, for transcripts of bench-mark elevations and for general remarks 
or explanation. 

In columns 2 and 6 write next above the red lines the continuous sums 
of the stadia intervals for the stretch. The sum of the last pair of con¬ 
tinuous sums in columns 2 and 6, divided by 0.002, will be equal to the 
distance in feet for the page; its equivalent in miles and tenths can be ob¬ 
tained from table in back of book. The total mileage from beginning of 
section on precise and of line on primary work must be given at bottom of 
each right-hand page. 

On primary work the algebraic sum of the page excesses for each day 
of backsights over foresights, with letter B if plus and F if minus, should 
be written in lower right-hand corner of right-hand page. 

The sum of readings in column 1 should be equal to those in 3, and the 
sum in column 5 should equal that in 7, and each should be written at the 
bottom of the page. Their difference should be written at the bottom 
of Column 4, and this should equal the difference obtained by subtracting 
the first from the last elevations in column 4. 

The formula 3C (Col. 2-C0I. 6), etc., at the bottom of the right-hand 
page, is for computing the correction to the elevations for errors of adjust¬ 
ment. This computation need not be made in the field. C is the constant 
which results from the “peg method” adjustment. By (Col. 2-C0I 6) is 
meant the difference of the “continuous sums of the stadia intervals of 
columns 2 and 6. 

Notes should be kept in ink and under no circumstances should erasures 
be made, a single line being drawn through erroneous records. 

Levelmen must number their note books consecutively, using numbers 
that do not conflict with those used by other levelmen for the same locality. 

144. Methods of Running.— In precise leveling a double 
line is is invariably run for the purpose of check on every 
bench-mark. The U- S. Engineers and the new Coast Survey 



334 


LEVELING OF PRECISION. 


methods adopt a sequence which is that already described for 
double rod for ordinary spirit-levels (Art. 130)* For speed 
they use two rodmen, and the levelman backsights on rodman 

_*-fW-* 

p 7 |\ A 


— 

Fig. 104. 


-SlNGLE-RODDING WITH TWO RODMEN. 


A at a x (Fig. 104) and foresights on rodman B at b 9 . Then 
the levelman and A move forward, and the former backsights 
on B at b % and foresights on A at a K . This is a single line of 
levels, and the party duplicate their own work by rerunning 
over the same line in an opposite direction each day. 

In the old Coast Survey method the levelman backsights on 
rodman A (Fig. 105) at the turning-point a x , and then back¬ 



sights on rodman B at the turning-point b 9 . Both A and B 
then pass him, and he then foresights on rodman A at turn¬ 
ing-point a 3 and on rodman B at b x , the rear turning-points 
a, and b 2 being left in the ground until the turning-points a % 
and b. are set. 

4 

145. Precise Rods. — The Coast Survey rod is of 
thoroughly paraffined wood, and the bottom, which is hemi¬ 
spherical, is set in saucer-shaped turning-points, the curvature 
of which is greater than that of the rod foot. This rod is 
single and non-extensible, 12 feet long, and divided into 
fractions of a meter by large, easily legible markings. At 
• short intervals on its face are inserted in the pine wood metal 
\plugs on each of which is engraved a fine line, and these are 
the zero marks on which the vernier is read; it being believed 
that these lines are finer than divisions can possibly be made 
upon wood. The rod can be read directly to thousandths of 
a meter, and by estimation to one ten-thousandth of a meter. 




PRECl E RODS. 


■7 7 

o 


Elevation.. 



Section. 



Fig. 106.—U. S. Geological Survey Double-target Level-rod 

One-third size. 


lo 






























































































LEVELING OF PRECISION. 




6 


The U. S. Engineers use a rod made of one piece of 
wood 12 feet in length. It has a T-shaped cross-section, 
a foot-plate, and a turning-point similar to the above. 
The rod is self-reading, that is, without targets, and 
graduated to centimeters. Closer records are made by 
estimation by the levelman, since there are three hori¬ 
zontal cross-wires in the instrument, on each of which 
readings are made, and the mean of these is the value 
used. 

The precise rods used by the U. S. Geological Survey 
are of two kinds, target-rods and speaking-rods. The 
double-tar get rods , the use of which is now abandoned 
(Fig. 106), was of the best selected white pine, well sea¬ 
soned and heated to a high temperature, when they are 
impregnated with boiling paraffine to a depth of one- 
eighth inch. The rods are a little over io feet long, and 
the graduations are commenced about a foot from the 
bottom of the rod to prevent readings being taken too 
near the bottom of the rod because of refraction. 

The single-target rods used are similar in all essential 
respects to the double rods just described, but have only 
one face divided and one target. They lack, therefore, 
the advantages gained by speed in manipulation with 
the double rods. They are, however, superior in speed 
and accuracy to other forms of target-rods (Fig. 96). 

The precise speaking-rods now exclusively used by 
the U. S. Geological Survey are an adaptation of the non- 
extensible speaking-rods of the Coast Survey. They are 
made as above described for Coast Survey rods, except¬ 
ing that they are divided into four yards instead of 
meters and can be read direct to thousandths of a yard 
(Fig. 107). 

146. Manipulation of Instrument. —In precise level¬ 
ing several important details of manipulation, although 

Fig. 107. —Precise Leveling-rods, U. S. Geological Survey. 









LENGTH OF SIGHT. 


337 


apparently trivial, add greatly to the accuracy of the resu’t. In 
addition to the necessity of exactly equalizing sights and of taking 
care not to refocus the instrument without adjustment, care 
should be taken to loosen the instrument from the tripod by 
freeing the central holding-screw after the tripod has been 
firmly planted in the ground, The instrument then rests on 
the tripod merely by its own weight and is not subject to the 
torsional strain which may be brought upon it by the tension 
of the center holding-screw. The three screws which bind 
the wooden tripod legs to the metal tripod head should be 
loosened after the tripod has been firmly planted, and then 
ret ightened before the observations are made, so as to obviate 
strain in the tripod and its head due to any twist brought 
against these screws in planting the tripod. After giving the 
final signal to clamp the target, the instrumentman should 
have the rod replaced on the turning-point, should again 
notice the level-bubble, and take a last look at the target 
bisection, calling out to the rodman, “ plumb.” or some sim¬ 
ilar word, at the moment the same is repeated by the iodman, 
so as to make sure that the rod is plumb at the moment 
of target bisection and after the target has been clamped. 

147. Length of Sight.—There is a limit of distance at 
which the rod should be placed from the instrument, which 
is variable and is dependent chiefly upon— 

1. Magnifying power of the telescope; 

2. Quality of work being done; 

3. Atmospheric conditions; and 

4. Sensitiveness of the level bubble. 

The first condition affects both the nearness and the 
extreme distance at which sights should be taken. If the rod 
is too close to the instrument, difficulty will be experienced 
in properly setting the target or bisecting the divisions of the 
rod if the latter is self-reading, and the levelman may waste 
much time in an effort to find too close a reading. There is 
also sometimes difficulty in focusing on a very near rod, but 


333 


LEVELING OF PFECISION. 


above all is the slowness caused by the short sights. Effort 
should therefore be made to take as long sights as are per¬ 
missible. Accordingly, the four classes of limitations above 
specified may be all taken to limit the greatest length of 
sight rather than the least. The distance of the rod from the 
instrument should not be so far that the magnifying power 
of the telescope will not permit of reading the rod or setting 
the target to the smallest division of the rod. 

The second limitation to distance, the quality of the work , 
gives the greatest latitude in distance of sight. If rough or 
flying levels are being run and only turning-points taken, and 
these as far apart as the power of the instrument will permit, 
or if the rod is being read to the .01 or even . 1 of a foot for 
the obtaining of approximate elevations only, the rod may 
be placed at as great a distance as the target or the divisions 
upon the rod are clearly visible, providing, of course, that the 
greater the distance of the rod from the instrument the more 
nearly the foresights and backsights should be equalized, 
otherwise errors will be introduced into the work owing to 
the errors in the bubble and the instrument adjustments. 

The third limitation to distance, atmospheric conditions, is 
one of the most important, since, when the atmosphere is 
vibrating rapidly because of heat, the difficulties of accurately 
reading the rod or bisecting the target become so great as 
to render it impossible to make the observations within the 
limit of a rod division, the cross-hairs of the instrument fre¬ 
quently dancing over several thousandths or even hundredths 
of a foot on the rod if it is placed at a considerable distance. 
Accordingly, as heat vibrations increase, the lengths of the 
sights must be diminished; and it is not uncommon, in very 
accurate work, to have to reduce sights to as low as 100 feet, 
and even then the results of a rod setting may be in doubt. 
Precise leveling should not be carried on in very hot weather 
or when the atmosphere is vibrating violently from heat or 
other causes. 


SOURCES OF EE FOE. 


339 


Atmospheric conditions, the magnifying power of the 
glasses, and other elements being satisfactory, the true limit 
of distance is fixed by the sensitiveness of the bubble. For 
instance, with an 8-second bubble the target can be set with 
comparative certainty to within .001 of a foot at a distance 
of a little less than 300 feet. Likewise, with a 4-second 
bubble on the same instrument the target can be set to .001 
of a foot with comparative accuracy at a distance of about 
400 feet. Accordingly, these distances for the instrument 
under consideration practically fix the limits of distance at 
which the rod may be placed under favorable atmospheric 
and other conditions. The ordinary engineer’s level has a 
20-second bubble, one which therefore for accurate work 
would limit the distance even more greatly ; that is, with such 
an instrument rod readings of less than .01 of a foot are rarely 
possible with accuracy. The accuracy of the same instru¬ 
ment is greatly increased by use of a io-second bubble. It 
may be stated that, in ordinary engineering levels, sights as 
long as 300 to 500 feet may be regularly taken. In precise 
levels, however, 350 feet should not be exceeded even with 
an instrument having a 2-second bubble, for though the 
sensitiveness of the bubble is increased, the other functions 
of the instrumental error, atmosphere, magnifying power, 
etc., do not increase in equal ratio. 

148. Sources of Error. —The operation of spirit-leveling 
involves perhaps more varieties of errors than occur in the 
use of any other engineering instrument. Moreover, these 
are of such peculiar kinds as to involve a fine distinction 
between such as are compensating and such as are cumula¬ 
tive. The sources of error may be divided into— 

1. Instrumental errors; 

2. Atmospheric errors; 

3. Rod errors, including turning-point and record; and 

4. Errors of manipulation. 

Among instrumental errors the most important is perhaps 


340 


LEVELING OF PRECISION. 


that due to the line of sight not being parallel to the level- 
bubble, and may be caused by imperfect adjustment or 
unequal size of the rings or both. If the telescope-slide is 
not straight or does not fit well, it will introduce an error. 
All of these errors may be eliminated by placing the instru¬ 
ment midway between the turning-points, and wherever ac¬ 
curate results, as in precise leveling, are desired, the lengths 
of foresights and backsights should be exactly equalized. In 
precise work the error of the telescope-slide is practically 
eliminated by not changing the focus after adjustment of the 
instrument. This would necessitate readjusting the instru¬ 
ment if for any reason the lengths of sights should be 
changed in any part of the day’s run. Another source of 
error arising from the instrument is produced by the adhesion 
of the fluid inside the glass tube, which prevents the bubble 
from coming precisely to its true point of equilibrium. This 
frequently occurs owing to the crystallization of something 
which is contained in the ether, little granules or crystals 
forming on the inside of the glass which catch the bubble 
and keep it from running smoothly. Careful microscopic 
examination of the bubble tube may show these crystals, and 
if discovered it should be discarded. 

The most important of atmospheric errors is the effect of 
the heat of the sun on one end of the telescope raising it by 
unequal expansion. This error may be partially eliminated 
in ordinary leveling by rapid manipulation of the instrument, 
so as to leave the least interval in which the sun may act. 
The error is greatest in work towards or from the sun and is 
cumulative; for if on the backsight the Y nearer the object 
glass is expanded, thus elevating the line of sight, then the 
other Y is expanded in the foresight, thus depressing the line 
of sight. This is a much greater source of error than is 
ordinarily recognized, for the error in the case above cited 
is further increased on the foresight by the cooling of the Y, 
which is expanded on the backsight. The sources of error 


* 


SOURCES OF FEE OF. 341 

due to this cause may be largely eliminated by shading the 
instrument from the sun, and this should be done in careful 
engineering as well as in precise leveling. 

Another class of atmospheric error is due to the jarring 
or shaking both of the instrument and of the rod by high 
winds. When the wind has become so high that in looking 
through the telescope the cross-hairs dance to such an extent 
as to prevent accurately sighting the target; or when it is evi¬ 
dent that the jarring of the instrument interferes with the 
exact leveling of the bubble; or when the rod itself vibrates 
to such an extent as to make it impracticable to exactly 
sight it by the instrument, precise leveling observations 
should be discontinued. The effect of high winds may be 
partially obviated by using fine wires or cords held by men 
to guy the top of the rod, and they may be obviated in the 
instrument by screening it either with an umbrella, wind¬ 
break, or a tent. In precise leveling by the Coast Survey on 
the plains of Nebraska, the wind has been so high continu¬ 
ously for weeks at a time as to render it necessary even to 
work in a high wind, and the harmful effect of the latter has 
been neutralized by guying the rods and by erecting a 
shelter-tent at every sighting. In running along the line of 
the Union Pacific Railroad a shelter-tent was carried on a 
frame on a hand-car in such manner that the instrument 
could be set up on the ground under the tent, and thus 
scarcely any time was lost in the operation. 

A most serious atmospheric error is that due to frost , or 
especially a frost following rain or melting snow. The writer 
has observed instances where tripod legs, firmly inserted in 
the frozen ground in the morning, when the sun was causing 
rapid thawing, have in the course of a few minutes—in fact, 
during the time the instrument was being sighted after level¬ 
ing—sunk so quickly as to keep the bubble continuously in 
motion, thus rendering it impossible to get a stationary posi¬ 
tion of the bubble. This was due to the heat of the metal 


342 


LEVELING OF PRECISION. 


tips of the tripod, warmed while the instrument was car¬ 
ried in the air, thawing the surrounding frozen ground, the 
water from which acted as a lubricant and permitted the 
tripod to sink. Precise leveling should not be conducted 
under such circumstances; for not only is the instrument 
affected, but also the turning-points on which the rod rests 
are liable to some movement, however carefully made and 
placed. The effects of dancing of the air and of refraction 
are referred to in Articles ill and 152. 

Rod and turning-point errors are of the same kind. 
Among the latter is error due to settlement or jarring of the 
turning-point or to its inferior quality. ^The first of these is to 
be guarded against only by using tops of railroad rails as rod 
supports or steel turning-points and driving them firmly into the 
ground with a heavy hand-sledge; and by care in placing the 
rod on the point so as not to produce any impact; and by 
carefully wiping the bottom of the rod and top of the turnings 
point prior to each setting. Errors of rod reading are to be 
guarded against by the levelman reading the rod and record¬ 
ing it himself when he and the rodman pass, so as to get a 
check on the reading of the rod by the latter, also in dupli¬ 
cate rodding by the two rods being read by the two rodmen 
as well as by the levelman. 

Lack of verticality of rod is to be remedied by waving it 
slowly backward and forward that the instrumentman may see 
that the cross-hair is tangent to a rod graduation at its highest 
point; or, better, by the use of rod levels, two of which 
are attached at right angles to the side of the rod, though a 
single circular level may be employed. In the use of these 
levels, that which determines the verticality of the rod later¬ 
ally scarcely need be noted by the rodman, as the vertical 
cross-hair of the spirit-level determines it in that direction. 
Another source of error in rods is due to inaccurate grad¬ 
uation. When done by a first-class instrument-maker and 
tested by the standards which he has in his possession, this 


DIVERGENCE OF DUPLICATE LEVEL LINES. 343 

source of error is generally found to be very small, yet for 
precise leveling the graduation should be tested by means 
of an official standard, and the error, however small, recorded 
and applied to each rod reading. Changes in rod length due 
to variation in temperature and moisture are so small that 
they may be disregarded in rods made of the best quality 
of well-seasoned white pine treated with paraffine as described 
in Article 145. 

149. Divergence of Duplicate Level Lines.—A curious 
fact, probably first noted in the United States in the report of 
the Chief of Engineers of the Army for 1884, but since fre¬ 
quently observed by the U. S. Coast and Geodetic Survey, the 
U. S. Geological Survey, and others doing precise leveling, 
is the fact that when duplicate lines are run, either in opposite 
directions by two sets of levelers or by the use of a single 
instrument reading on two rods, the discrepancies between 
the two lines have an average tendency in one direction or to 
one sign, and increase with the distance. In other words, the 
tzvo lines separate as they progress, the distance between the 
heights of any fixed bench-mark as determined by them in¬ 
creasing with the length of the line. Many reasons have been 
assigned for this, as settlement of instrument or of turning- 
points, effect of sun, illumination of target, frost, etc., but 
scarcely any are quite satisfactory. Remedies have been sug¬ 
gested, such as leveling alternate sections in opposite directions, 
or reading the backsight first at each alternate setting of the 
instrument, but no complete remedy has been yet discovered. 

The writer’s experience with such work on the Geological 
Survey indicates that the best results are obtained by a dupli¬ 
cate rodded line (Art. 143), or by running two lines in opposite 
directions or in alternate sections. He believes that these 
errors are in some measure due to settlement of the instru¬ 
ment between the time of taking back- and foresights and be¬ 
tween the time of observing on the two separate lines or rods. 
With the aid of Mr. W. Carvel Hall of the Geological Sur- 


344 


LEVELING OE PRECISION. 


vey he has reduced this form of error to a minimum by quick 
manipulation ; by the employment of the method of rod suc¬ 
cession, whereby immediately after the backsight the foresight 
can be at once read (Art. 143), and by using double-faced rods 
(Art. 145), thus reducing the time consumed in reading the 
rod between the various sights. 

Recent experiments by the U. S. Coast and Geodetic Sur¬ 
vey in running precise levels clearly show that errors causing 
divergence of duplicate lines are produced in large measure by 
rising or subsidence of metal turning-points driven in the 
ground. Also that these errors can be greatly reduced by 
using the tops of rails on railroads as turning-points. 

150. Limit of Precision.—The final error of a series of 
observations will , according to the theory of probabilities, 
vary as the square root of the number of observations when 
affected only by accidental errors. Accordingly, when the 
instrument is set up the same number of times per mile, the 
error of leveling a given distance is assumed to be in propor¬ 
tion to the square root of the distance, and not to vary directly 
as the distance. In fact, a limit of error based on this 
presumption, while found to be very satisfactory for short 
distances, say those under one hundred miles, proves too 
severe for greater distances, and it is almost impossible to 
maintain it for such great distances as are leveled over by 
lines of precision. This is probably true because accidental 
errors are not the only ones made, and the number of obser¬ 
vations are not solely proportional to the distance leveled, 
that is, the lengths of sight are not constant. While a fixed 
limit of precision may be maintained for a number of short 
pieces of leveling, it will generally be exceeded if the sums of 
errors be added together as the total discrepancy. 

Various limits of precision have been fixed in accord¬ 
ance with the theory of probabilities by different precise- 
leveling surveys. If the probable error of leveling one mile 
be e' , then that for leveling d miles is e — e f Vd. Levels 


ADJUSTMENT OF GROUP OF LEVEL CIRCUITS. 345 

of precision executed in Europe of late years show that 
the probable error of level lines of precision should not 

exceed 5 mm. ^distance in kilometers, equivalent to about 

.021 ft. V distance in miles, the result being in feet. The 
U. S. Coast and Geodetic Survey calls for a precision in feet 

equivalent to .02 ft. V distance in miles; the British Ord¬ 
nance Survey endeavors to place a high limit in fixing a 
constant error of 0.01 foot per mile, and yet this same limit 
applied to any of the long lines of precision run in the United 
States is very much easier to attain than any of the limits 
fixed above, because it varies directly as the distance. 

The U. S. Geological Survey has fixed as its limits of 
precision in its precise leveling that of the Coast Survey, 

namely, a result in feet = 0.02 ft. ^distance in miles, or 

= .02 ft. V2d miles for duplicate lines. The U. S. Missis¬ 
sippi and Missouri River Commissions aim at a limit repre¬ 
sented by the formula 0.0126 ft. V2 X distance in miles for 
direct lines. The maximum discrepancy now allowed by the 
Coast Survey between duplicate runnings of a mile long is 5 
mm. or one-fifth inch. 

151. Adjustment of Group of Level Circuits. —Where a 
line of levels has been run in such manner as to connect back 
on itself, thus forming a polygonal figure or circuit , there will 
occur some error of closure. If the instrument be set up 
the same number of times in one mile, the probable error of 
the result increases as the square root of the distance. In at¬ 
tempting to distribute the error in such a closed circuit it 
.must be remembered that the weights to he applied a? e in¬ 
versely proportional to the squares of the probable cirors , 01, 
in other words, to the distance over which the leveling is cai- 
ried. If the leveling be run over three routes, A, C, and D 
between the points A and B (Fig. 108) and the lengths of 
these be respectively 5, 7 -> and 8 miles > the weights to be 
applied to them will be respectively y, and J. 









346 


LEVELING OE PRECISION. 


If a closed circuit of levels is run from A via C, B , and D 
back to A, and bench-marks are set at each of those points, the 
adjusted elevations of these benches should be in direct pro¬ 
portion to the distances between the benches. If the distance 
from A to C is 4 miles, from C to B 3 miles, from B to D 3 
miles, and from the latter to A again 5 miles, then the total 
distance is 15 miles. Therefore^ of the total discrepancy 



is to be subtracted from the elevation of the first bench, C ; 

of the total discrepancy is to be subtracted from the 
second bench, B ; from the third bench, D , etc.,—account 
of course to be taken of signs. 

A group or net of levels such as that shown in Fig. 109 
permits of the computation of the elevations of the various 
bench-marks by several different routes. If now the eleva¬ 
tion of any one bench be given, the elevations of the other 
junction-points are to be obtained. The number of inde¬ 
pendent quantities in any such group of level circuits is one 
less than the number of connecting benches. If this group of 
levels be adjusted by the method of least squares, there will 
be introduced as many conditional equations as there are 
separate geometric figures and one less independent quantity 
than there are connecting bench-points. 

A simpler method of adjustment, however, that recom¬ 
mended by Prof. J. B. Johnson and preferred by the author, 
is to consider the errors in proportion to the square roots of 
the distances or lengths of the sides of the polygonal figures. 
This is because the errors are compensating in their nature 
and increase with the square roots of the lengths of the lines. 





REFRACTION AND CURVATURE. 


34 7 


Instead, therefore, of solving the group by least squares as 
one system, that polygonal figure having the largest error of 
closure should be first adjusted by distributing its error among 
its sides in proportion to the square roots of their length. 
Then the circuit or polygon having the next largest error 



Fig. 109.—Group of Connected Lf.vei. Circuits. 


should be similarly adjusted, using the new values for the 
adjusted side if contiguous to the former, and distributing the 
remaining error among the remaining sides of the figure with¬ 
out distributing the side already adjusted. 

152. Refraction and Curvature. —The line of sight of a 
telescope when the bubble is level is theoretically parallel to 
that of the surface of the ocean at rest. In fact, however, 
it is depressed below that plane by the action of refraction , 
and it lies between the level or curved surface of the ocean 











34 « 


LEVELING OF PRECISION. 


and a tangent plane to the same, but is nearer the latter. 
The deviation of the tangent plane from the level surface is 
about f of a foot per mile, and for n miles it is § n " 1 feet. 

In all spirit-leveling and trigonometric operations curva¬ 
ture and refraction are rarely considered separately, but are 
usually treated in combination (Art. 166). Their combined 
effect is to cause the line of sight to be elevated above the 
level of the surface by an amount equal to about 0.57 foot in 
one mile, or for n miles by 0.57;/ feet. The above facts, how¬ 
ever, have little bearing on the ordinary operations of spirit¬ 
leveling, as the lengths of the sights taken are too short to be 
affected appreciably by them. Moreover, so long as the rule 
is strictly adhered to that the lengths of backsights and fore¬ 
sights shall be equal, all effects due to curvature and refrac¬ 
tion will be eliminated. 

In long-distance leveling (Art. 155) the effects of curva¬ 
ture and refraction become immediately appreciable in amount 
and must be taken into consideration if sights are not equal¬ 
ized. Ordinarily, however, they are eliminated in this form 
of leveling by simultaneous reciprocal readings with two 
instruments, or ordinarily less accurately by frequently repeated 
reciprocal readings from either end, thus equalizing the lengths 
of the sights. 

One of the most abundant causes of error in leveling is 
the refraction encountered by the line of sight passing near 
to the surface of the earth, and also another phenomenon 
nearly related to it—the dancing of the air due to heat-waves 
near the ground surface. This latter can only be eliminated 
satisfactorily by reducing the length of the sight when the 
air is boiling badly. This reduction must be of such amount 
that the space on the rod danced over by the cross-hair will 
not be of appreciable amount. Refraction may be reduced 
to a minimum by exercising the precaution of never sighting 
too short a rod—that is, never allowing the line of sight 
to come nearer the ground than ii to 2 feet (Art. 111). 


SPEED AND COST OF LEVELING. 349 

This precaution should be especially observed at that time of 
day at which refraction is greatest. 

153. Speed in Leveling. —The speed with which levels 
can be run varies greatly with the accuracy desired, the 
character of the country, the atmospheric conditions, the 
method of running employed, and the levelmen and rodmen. 
In ordinary or flying levels, in which merely turning-points 
are taken and no great accuracy is aimed at and a self-reading 
rod employed, speeds of from 3 to 1 5 miles a working day are 
attainable, the lowest in very hilly country, the highest on 
comparatively flat plains. Engineering levels of considerable 
accuracy, such as the primary spirit-levels of the Geological 
Survey, are run at speeds varying under average conditions 
from 50 miles to 90 miles per month of about twenty work¬ 
ing days. 

Strange as it may seem, precise levels are run with a 
generally higher average speed than are the ordinary levels 
above cited. One reason is because they are invariably run 
over the best and most favorable grades, generally following 
the lines of railways. The chief reason is because they are 
run with two rodmen, so that no time is lost by the levelman 
or rodmen waiting for one another to move to the next posi¬ 
tion. The Coast Survey have run in recent years with 
their new level at speeds of from 5 to 9 miles a day, the greater 
speed being made under favorable atmospheric conditions. 
The precise levels of the Geological Survey were run with an 
average for the seasons 1896 to 1899 varying between 4 and 8 
miles per day as limits. Both organizations have averaged on 
runs over 1000 miles as high as 65 miles per month, with 
maximum of 120 miles per month. 

154. Cost of Leveling. — Necessarily the cost of leveling 
varies according to the character of the work. A party 
which is organized for a long season of work will operate less 
expensively than one which is placed in the field for but a 
short period of time. The following estimates are based on 
seasons of at least several months’ duration. 


350 


LEVELING OF PRECISION. 


Ordinary or flying levels run by the Geological Sur¬ 
vey along good roads in New England with a party consist¬ 
ing of levelman and rodman only, living on the country, 
average a cost of $2.50 per linear mile. The primary or 
engineering levels of the same organization run by a level- 
man and rodman only, but over all sorts of routes, since they 
are compelled to place a bench-mark once in every thirty- 
six square miles, and where subsistence is had either in hotels 
or farm-houses or in camp, vary in cost from $6.50 per 
linear mile in rough mountain country like the Adirondacks, 
West Virginia mountains, or Oregon, as one extreme, to 
$3.5° per linear mile in flat country like Alabama, western 
New York, and the Mississippi valley. 

Where the work is executed in the best manner, as above 
described, and the rod is set only on turning-points and not 
on intermediate stations, a fair estimate of the cost can be 
had from an inspection of Table XIII giving the result 
of the work done by the various leveling parties working 
in different States and under different climatic and topo¬ 
graphic conditions for the U. S. Geological Survey during 
the field season of 1896. The bench-marks enumerated were 


Table XIII. 

COST OF LEVELING PER MILE IN VARIOUS STATES 


State. 

Miles of Levels. 

Number of 
Bench-marks. 

Cost per Linear 
Mile. 

Alabama. .. 

65 

IO 

$4.30 

Arkansas. 

179 

15 

3-75 

California. 

338 

72 

11.27 

Colorado. 

404 

77 

5.80 

Delaware. 

40 

M 

2.80 

Georgia. 

278 

38 

4 - 3 ° 

Idaho. 

140 

25 

7-53 

Illinois. 

129 

7 


Indian Territory... 

4.174 

700 

.... 

Iowa. 

236 

43 

3-98 

Kansas. 

43 

15 


Maryland .. 

120 

20 

10 

00 

0 

Michigan. 

90 

6 

4 \ 5 ° 

Missouri. 

316 

35 

3 - 9 ° 

Montana. 

200 

29 

4 49 


State. 

Miles of Levels. 

Number of 
Bench-marks. 

Nebraska . 

365 

IOO 

New York . 

925 

105 

North Carolina ... 

597 

108 

North Dakota. 

76 

l6 

Oregon ... ., . 

130 

24 

South Dakota. 

320 

42 

Texas . 

1,098 

222 

Vermont. 

40 

8 

Washington . 

186 

40 

West Virginia. 

180 

35 

Wyoming. 

304 

58 

Totals and aver- 



age. 

10,968 

1,924 


CTj 

C 





O'. 

O 

U 


$2.85 

3.66 

415 

6-53 

3.26 

2. 79 

4-44 

3.80 

8.44 

4.44 
8.17 


$4-78 
















































SPEED AND COST OF LEVELING. 351 

of the permanent metal forms (Fig. ioo), and these added 
somewhat to the cost. Where less careful work is attempted, 
the cost may be reduced as much as one-half for each kind of 
country, and where intermediate stakes are set, say for every 
one hundred feet for railway leveling, the cost will be 
increased by at least one-half. With their new instrument the 
Coast Survey leveling of recent years has ranged from $7 to 
$11 per mile run. 

Precise leveling executed in connection with city surveys 
is necessarily more expensive and scarcely as accurate as that 
carried on elsewhere, because of the annoyance and jarring 
from passing vehicles, rapid alternation of sunshine and 
shadow about buildings, etc. In the precise leveling done in 
connection with the survey of the city of Baltimore, there 
were run 141 miles of double line, in the course of which 
there were established 606 permanent bench-marks, or one to 
every 1228 linear feet. As the area of the city survey was 
30 square miles, there were established 20 bench-marks per 
square mile. The computed probable error of the work was 
about 0.003 of a foot per mile, about the same being the 
probable error of the precise leveling in the city of St. Louis. 
The cost of precise leveling in the city of Baltimore for field 

Table XIV. 


COST AND SPEED OF GOVERNMENT PRECISE LEVELING. 


Organization. 

Year. 

Locality. 

Days of Actual 
Field-work. 

Miles of Dupli¬ 
cate Line. 

Total Cost. 

Speed. Miles 
per Day. 

Cost per Mile. 

Cost per Day. 

• 

Engineer Corps. 

1882 

Carrollton. La., to 









Biloxi, Miss. 

35 

87 

$2778 

2-5 

Is 1 -93 

$79.37 

«( 

1882 

Keokuk, la., to 









Fulton, Ill. 

50 

170 

3252 

3-4 

19.08 

65.04 

41 

1893 

Blair, Neb., to De- 









witt, Mo. 

22 

32 

736 

1-5 

23.00 

33 - 5 ° 

Coast Survey (old style) 

1895 

Richmond, Va., to 









Washington,D.C. 

55 

IX S | 





44 *4 

1895 

Lamar, Mo., to 


} 

3900 

2.0 

10.94 

31.20 



Chester, Ark. 

70 

150 ) 





Geological Survey .. 

1896 

Morehead City, N. 









C.,to Paint Rock, 









N. C. 

!05 

457 

2280 

4-3 

5.00 

O 

M 

Cl 

44 

1897 

Paint Rock, N. C., 









to Atlanta, Ga... 

48 

3°8 

1172 

6.4 

3-78 

24.40 

































352 


LEVELING OF PRECISION . 


and office work averaged $23.56 per mile, that for the city of 
St. Louis averaging $45.38 per mile. 

155. Long-distance Precise Leveling. —In running pre¬ 
cise levels it may occur that, owing to unusual physical condi¬ 
tions, the line cannot be carried forward by short and equal 
foresights and backsights, as in crossing an expanse of water. 
Under such circumstances, long sights, involving special 
methods of observation and reduction, become necessary. In 
long-distance leveling, in order to attain the accuracy of pre¬ 
cise leveling, instrumental and atmospheric errors are elimi¬ 
nated by taking simultaneous reciprocal observations. 

The instrument employed should be a good precise level, 
and the rods should be provided with large targets up to 12 
inches square for distances of two miles. The target should 
be painted one color, preferably red, with a white band across 
its center, one to two inches wide at the outer edge of the 
target and narrowing to -§ inch wide at the opening in the target 
center, this white streak to be bisected by the cross-hairs, and 
provided with a cross-wire opposite its center for convenience 
in target reading. The instruments and rods should rest on 
solid foundations; and in leveling across water, the more usual 
case in which such work is done, the telescope should be 10 
to 15 feet above the water surface to avoid extreme refrac¬ 
tion. The instrument should rest on a platform independent 
from any surrounding platform on which the observer may 
stand. 

In such a piece of work conducted by Mr. Gerald Bagnall 
for the U. S. Engineer Corps at Galveston, Texas, platforms 
had to be erected in the water, and owing to the unstable 
character of the bottom an apron of rubble was placed around 
them. The corners and supports for the instruments were 
heavy piles driven 16 feet into the bottom, well braced hori¬ 
zontally and diagonally. Rocks were placed around the out¬ 
side piles, and rows of sheet-piling were driven along them. 
A reference bench-mark was placed near each instrument and 


LONG-DISTANCE PRECISE LEVELING. 


353 


nearly at right angles to the directions of the line joining the 
two instruments, so that the long sight of both observers 
might be equal. Each leveling party consisted of an observer, 
a recorder, rodman, umbrellaman with the instrument, and an 
assistant to signal and watch signals with the glasses. Simulta¬ 
neous reciprocal observations were taken with the two instru¬ 
ments, one at each end of the line, in order to eliminate the 
error due to refraction, and this was effected by signaling be¬ 
tween the two so that the targets were set at the same moment. 

The errors which have to be eliminated by this system are: 

1. Those due to the inclination of the bubble and to collima- 
tion, which are eliminated by each instrument independently. 

2. Those due to curvature and refraction, which are 
eliminated by the simultaneous reciprocal observations. 

3. Those due to the inequalities of pivot-rings of both 
telescopes, which are eliminated by the observers changing 
stations and repeating the observations. 

A set of observations at each station should consist of at least 
four rod readings taken with telescope and level direct and re¬ 
versed, thus: 1st, telescope and level direct; 2d, telescope direct 
and level reversed ; 3d, telescope inverted and level reversed ; 
4th, telescope inverted and level direct. When a sufficient 
number of sets of observations have been taken the observers 
change stations and repeat the operation, determining the 
true difference of elevation of the reference bench-marks, from 
which the heights of instruments for the long sights are deter¬ 
mined. The maximum distance at which satisfactory results 
may be obtained depends on the instrument used and the con¬ 
ditions surrounding the work. With an instrument having a 
powerful object-glass, and high magnifying power being used, 
fair results may be obtained at distances up to two miles. 

Another example of long-distance leveling is given here 
from the observations from one of three days in which the 
precise levels of the U. S. Geological Survey were carried 
across the Tennessee River by Mr. W. Carvel Hall, the 


354 


LEVELING OF PRECISION. 


greatest length of sight being 1810 ft. (Fig. IIO.) In this 
work a 4-second bubble was used and a 40-diameter magni¬ 
fying power. Lozenge-shaped pieces of paper, 0.07 ft. in 
width, were placed on the targets as markers. The sights 
were 85 ft. clear above the river surface. Two reference 
points, A and D , were placed on the near bank at distances 
of 15 and 20 ft. respectively, as backsights in the forward 
crossing, these being terminal points in the regular line. On 
the far bank of the river two other reference points were 



Fig. no. —Long-distance Leveling across Tennessee River. 


placed, both at a distance of 48 ft. beyond the instrument, 
but at some little distance apart one from the other, and these 
became rear turning-points in the continuation of the regular 
line. The foresights taken from the near bank on the two 
lines, after backsighting on the reference points, were respec¬ 
tively 1785 and 1790 ft. in length, and the backsights taken 
from the far bank to the rods on the near bank, which were 
placed on the reference marks as turning-points, were respec¬ 
tively 1805 and 1810 ft., or sufficiently close to the length of 
the foresights to practically eliminate errors due to curvature. 
The errors due to refraction were eliminated as far as possi¬ 
ble by observing at such times in the day as refraction was 
least, namely, late in the morning, and when the day was 
slightly cloudy, the atmosphere still, and there was no per¬ 
ceptible “boiling” of the air; also by observing at three 
different times on another day under different atmospheric 
conditions. 

From the instrument position 1 on the near bank a read- 















LONG-DISTANCE PRECISE LEVELING. 


355 


ing was first made on rear reference point A , and then ten 
readings were made on fore reference point B on the other 
side of the river; then a reading was made on reference point 
D on the second line, and ten readings were made on the dis¬ 
tant point C across the river. Likewise, from instrument 
position 2 on the far bank one reading was made on reference 
point B, and ten on the distant back turning-point A on the 
rear bank; also one on the near reference point C, and ten on 
the distant reference point D on the rear bank. The follow¬ 
ing are the results of the four sets of observations: 


5-301 

6.076 


4.295 

5 ‘ 3 I 9 

6.096 

4.219 

4.293 

5-312 

6.109 

4.233 

4.288 

5-329 

6.108 

4.272 

4.299 

5-324 

6.094 

4.242 

4.307 

5-318 

6.103 

4.249 

4.300 

5-338 

6.109 

4.225 

4.304 

5-301 

6.105 

4.247 

4.292 

5-317 

6.091 

4.245 

4.282 

5 - 3 I 9 

6.098 

4.224 

4-3 11 

Means: 5.318 

6.099 

4.237 

4.297 

The resulting elevations 

of the two 

turning-points on the 


far bank, as obtained from the above observations, were, in 
feet: 

Turning-point 571 + 3675 : from east bank, 807.2 11 ; from 
west bank, 807.203; mean, 807.207; extreme difference of 
elevation, 0.008. 

Turning-point 571 + 3670: from east bank, 805.523 ; from 
west bank, 805.514; mean, 805.518; extreme difference of 
elevation, 0.009. 

The divergence of the lines for this day’s work was: at the 
east bank, 0.911 ft. ; at the west bank, 0.927 ft. 

156. Hand-levels. —A very useful little instrument for 
the topographer is the hand-level, by which approximate level 







356 


LEVELING OF PRECISION. 


lines can be determined for some distance from the position 
of the observer and thus aid him in following the course of 
level or contour lines. This instrument consists of a brass 
tube six inches in length with a small level on top near the 
object end. (Fig. m.) Beneath is an opening through which 
the bubble can be seen as reflected from a prism into the eye 
at one end. Both ends are covered by plain glass, while there 
is a small semi-convex lens in the eye end to magnify the level- 
bubble and the cross-wires beneath the bubble. The cross¬ 



wires are fastened to a small frame moving under the bubble- 
tube, and are adjusted in place by a small screw at the end of 
the bubble-case. By standing erect and sighting any object 
and lowering or raising the object end of the level by hand 
until the reflection of the bubble is exactly bisected by the 
cross-wires, a horizontal line will then be sighted and the po¬ 
sition of the horizontal cross-wire will indicate approximately 
the elevation of any object which is at the same height as the 
eye of the observer. 

157. Using the Locke Hand-level. —There are two ways 

of leveling with the Locke hand-level. One is for the ob¬ 
server to stand erect, measure the height of his eye against 
a pole and note this height—say five feet. Then he directs 
the hand-level at the side of a hill or of a tree-trunk and 
notes where the horizontal wire intersects this. Then this 
object is at exactly the height of his eye above the ground, 
or five feet. Moving forward to it and standing with his 
feet on a level with this object, he is raised five feet, and, 
continuing the process, he levels along differences of five feet 
in elevation at a time. 





























ABNEY CLINOMETER LEVEL. 


357 


In sketching contours the hand-level is used differently. 
Standing on the ground and knowing his elevation, he adds 
to that the height of his eye. Then sighting along the slopes 
of the land with the Locke level, he observes where the hori¬ 
zontal line strikes the hillsides, and knows that such points are 
on a level with his eye, or five feet above the contour on which 
he stands, and he is thus able to sketch that contour with a 
considerable degree of accuracy. 

The topographer can with the Locke level determine the 
elevations of points about him which are but a little above or 
below his height, by sighting them and estimating the dis¬ 
tance above or below the level line as indicated by the cross¬ 
hair. If the points are at any considerable distance, he must 
make allowance for curvature and refraction. Great reliance 
must not be placed, however, on the accuracy of this instru¬ 
ment, as its results are but approximate. 

158. Abney Clinometer Level. —This is but an English 
modification of the Locke level, and is most useful in estimat¬ 
ing the angles of slope, or grades, and thus in sketching con- 



Fig. 112.—Abney Clinometer Hand-level. 


tours. It is also useful in reading rough vertical angles. 
Where a traverse plane-table (Art. 61) is used, however, it is 
more accurately replaced by a vertical angle sight-alidade 
(Art. 62). Attached to a hand-level is a small telescope 
revolving about a vertical arc graduated to 60 degrees on 












353 


LEVELING OF PRECISION. 


either side of zero when the instrument is held level (Fig. 
112). It can be used as the Locke level, and also with 
considerable accuracy by resting the tube, which is square, 
on a plane-table board or other surface which can be leveled. 
Having leveled the tube by holding it in the hand or resting 
it on a plane-table and noting that it is level by bringing the 
bubble against the horizontal cross-hair, the small telescope 
is then directed up the slope and the angle of the slope read; 
or it is directed at some object the distance of which is 
known, and with the angle read the difference in height can 
be computed (Art. 160). 


CHAPTER XVII. 


TRIGONOMETRIC LEVELING. 

159. Trigonometric Leveling. —Trigonometric leveling 
is the process of determining the difference in elevation be- 
tween two points by means of the angle measured at one of 
them between the horizontal or level line and the other; or 
by measuring the zenith distance of the other. This method 
of leveling is especially suited to finding the heights of sta¬ 
tions in a triangulation survey, and in connection with stadia 
traverse. In triangulation the vertical angles are measured 
with the same instrument as are the horizontal angles. In 
stadia and odometer traverse the vertical angles are measured 
with the same instrument and at the same time as is the 
distance or the deflection angle. 

Trigonometric leveling is primary or secondary in quality, 
depending upon the instruments and methods employed. In 
either a vertical angle is observed to the point the height of 
which is to be determined, and this, with the distance between 
the occupied and the observed points, gives the quantities 
necessary to determine their difference in elevation. Primary 
trigonometric leveling is performed by measuring at one sta¬ 
tion, with the vertical circle of a large theodolite (Art. 241), 
the double zenith distance (Art. 297) of the signal at the other 
station ; or by the measurement, by means of a micrometer 
inserted in the eyepiece of the telescope (Art. 242), of the 
differences in altitude between different stations, in connec- 

359 



360 


TRIGONOMETRIC LEVELING. 


tion with a reference mark the absolute height of which, or 
its zenith distance, has been previously obtained. Secondary 
trigonometrical leveling , or, as commonly called, vertical angu¬ 
lation , is performed with a small theodolite or with a tele¬ 
scopic alidade (Art. 59), and consists of direct measurement 
of the angle between stations observed and the horizon, as 
the latter is determined by the level-bubble on the instru¬ 
ment. A similar series of observations is taken at each suc¬ 
cessive station, and if the elevation of one of these is known 
the elevations of the others can be computed. 

In the process of trigonometric leveling, the height of the 
telescope above ground and the height of the signal must be 
carefully measured and made a part of the record, also the 
hour of making the observation, as in accurate work this has a 
bearing upon the correction for refraction. In trigonometric 
leveling of primary order the state of the level at the com¬ 
mencement and end of the observation, and observations 
made to determine value and sequence of arc corresponding to 
a turn of the micrometer-screw, become a part of the record, 
as does also the object sighted. 

The best results are obtained by measuring reciprocal 
zenith distances at two stations at the same moment of time, 
in which case the conditions of atmosphere are practically the 
same and the effects of refraction are eliminated. When re¬ 
ciprocal zenith distances are measured, not simultaneously 
but by the same observer on different dates, these should be 
made on various days from each station in order to obtain as 
far as possible a mean value of the angle and an average value 
of the refraction. The relative refraction (Art. 166) may be so 
different between various stations at distances greater than 15 
or 20 miles apart as to seriously affect the results unless a very 
large number of measures are taken on numerous and favorable 
days. The higher the elevation at which observations are 
made the more reliable the results; also, the larger the num¬ 
ber of stations included in a scheme of vertical triangulation 


VER TIC A L A NG ULA TION. 361 

the better the results, owing to the possibility of the adjust¬ 
ment of the whole. 

The results obtained by trigonometric leveling are of far 
greater accuracy than ordinarily supposed. The best work of 
this kind is that executed by the U. S. Coast and Geodetic 
Survey in connection with its transcontinental belt of primary 
triangulation. Checks on these levels have been obtained by 
means of precise spirit-levels to some of the triangulation 
stations. At. St. Albans base near Charleston, W. Va., the 
elevation by triangulation brought from the Atlantic coast is 
594.78 feet. The elevation of the same point by precise 
spirit-levels from Sandy Hook via Chillicothe is 595.616 feet, 
a difference of only 0.836 feet, which is much better than 
could be expected from spirit-levels of less accuracy than 
precise quality would produce. 

160. Vertical Angulation. —This term is used to desig¬ 
nate the process of obtaining elevations by angular methods 
of ordinary quality, as by telescopic alidade used with plane- 
table or by the vertical circle of a transit instrument. In 
this work the distances and angles are measured with only 
approximate accuracy because of the qualities of the instru¬ 
ments employed, the signal sights had are not clearly defined 
and accordingly corrections for curvature and refraction 
(Art. 166) are made but approximately. Instead of having 
a vertical arc which can be set at zero when the level- 
bubble attached to the telescope is leveled, it is better to 
record an index error and correct the angle for this. Thus 
the telescope is made level by the bubble, and the reading on 
the vernier is recorded under the title index error. Then the 
cross-hairs are directed to the object the elevations of which 
are to be determined, and the vernier is again read. The 
difference between the two readings gives the angle between 
the object sighted and the horizon, and is recorded in the 
notes as plus or minus. To apply the correction to vertical 
arc to the vertical angle attention must be paid to the signs; 



362 


TRIG ONOMEI 'RIC LE VELING. 


for a plus error in vertical arc subtract the error from plus 
angles and add to minus angles. 

An example of the mode of keeping such notes is as 
follows: 


Station, XXIII. Elevation, 2960'. 

Date , Nov. 16, 1898. 


Description of Point Sighted. 

No. 

Point. 

Level. 

Angle. 

Dist. 

Miles. 

Difif. 

Elev. 

Elev. 

Adj. 

Elev. 

Kitty Cobble; top. 

XV 

O / 

14 09 

O / 

13 IO 

O 

+ O 

/ 

59 

2.93 

270' 

3236 

3239 

Wolf Lake house; base. 

i 5-3 

IO 02 

14 °3 

- 4 

OI 

1.02 

377 

2583 

Top of ledge over Brooktrout 

17-9 

*3 07 

14 12 

~ I 

05 

i -43 

138 

2822 

2824 

Russia; cupola red barn. 

I 

13 00 

14 12 

— I 

12 

1.01 

no 

2850 



In vertical angulation corrections to the observed angles 
must be made for curvature and refraction (Art. 166), which 
may be taken from tables (Tables XVI and XXXI), also for 
the height of the instrument above ground surface and the 
height of signal. (See Table XV and example Art. 163, also 
Art. 239.) The correction for difference between the heights 
of signal and instrument above ground may be computed by 
the formula 

Cor. = — 7 ,.(27) 

cL sin r 7 

in which d is the distance between stations, and h the differ¬ 
ence in height (Art. 239.) Or, with fair approximation, a 
correction may be made by determining the differences in ele¬ 
vation observed, adding to the known height of the occupied 
station the height of the telescope above it before making 
the computations, and subtracting from the result or com¬ 
puted elevation of the station sighted at, the height of the 
target above ground. 

To sight the telescope on visible points of equal elevation 
the correction for curvature and refraction must be applied to 
the vernier reading, that is, the vernier must not be set at 
























VERTICAL ANGULATION , COMPUTATION. 363 

zero, but at a minus angle the number of minutes of which is 
nearly three-eighths of the distance in miles. 

161. Vertical Angulation, Computation. —The quantity 
entered in the distance column above is measured directly on 
the plane-table board or on the map. In the number column 
is the number of the station corresponding to the summit 
sighted if it has been occupied already; or if the point has 
been sighted from some other station, the number of the 
pointing which was given from that station; or if it has never 
been sighted for the other station, it is given a new number 
for the occupied station. Under the columns point and level 
are placed the angles read when the instrument is pointed at 
the object and when the telescope is leveled, providing it is an 
instrument which has not an adjustable vernier. In the col¬ 
umn difference of elevation is placed a quantity either com¬ 
puted (Art. 164) or taken from a simple table. 

Table XV is one which can be used for determining 
angles of elevation or depression up to any distance. For the 
first angle, for instance, take out 59' in the first column of 
the table. In the second column, that headed o°, the differ¬ 
ence of height is found corresponding to the unit distance one 
mile, and this is 90.6. This quantity multiplied by the dis¬ 
tance in miles, 2.93, gives a difference of elevation of 265.5 
feet. The correction to curvature and refraction for 2.93 
miles is 4.8 feet, which is always additive. As the angle in this 
case is positive the total difference of elevation is 270.3 feet. 

For all distances less than 1.6 miles the correction to cur¬ 
vature and refraction may be taken as 5 feet, as the height of 
instrument, about 4.5 feet, has to be added. 

Under the column adjusted elevation, in the above exam¬ 
ple, is given the final height of the point as obtained by aver¬ 
aging its elevation as determined from several stations. 

162. Vertical Angulation in Sketching.— The elevations 
o ( positions occupied by the topographer while sketching 
(Arts. 13 and 17) may be checked in practically the same 


364 


TRIGONOMETRIC LEVELING. 


TABLE XV.—DIFFERENCES OF ALTITUDE FROM 


Difference of altitude = 

j -f- D / t l -f h 2 for angles of elevation. 

1 — D / i l -j- h , 2 for angles of depression. 


0° 
h 1 

1° 

hx 

2° 

3° 

4° 

*1 

5° 

* 1 

6° 

*1 

7° 

hi 

/ 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

O 

0.0 

92.2 

184.4 

, 276.7 

369.3 

461.9 

555 -o 

648.3 

1 

*■5 

93 7 

185.9 

278.2 

370.7 

463-5 

556.5 

649.9 

2 

3 -1 

95-2 

187.4 

279.8 

372 3 

465-0 

558.0 

651.4 

3 

4.6 

96.8 

189.0 

281.3 

373-8 

466.6 

559-6 

653.0 

4 

6.1 

98-3 

190.5 

282.9 

375-4 

468.1 

561.2 

654-5 

.5 

7-7 . 

99.8 

192. I 

284.4 

376-9 

469.7 

562.7 

656.1 

6 

9-2 

101.4 

193.6 

286.0 

378.5 

• 471.2 

564-3 

657.7 

7 

10.7 

102.9 

195.1 

287-5 

380 0 

472.8 

565-8 

659.2 

8 

12.3 

IO4.4 

196.7 

289.0 

381.6 

474-3 

567.4 

660.8 

9 

13.8 

106.0 

198.2 

290.6 

383-1 

475-9 

568.9 

662.3 

10 

15-4 

i° 7-5 

199.8 

292.1 

384.6 

477-4 

570-5 

663.9 

11 

16.9 

IO9. I 

201.3 

293.7 

386 2 

479.0 

572.0 

665-5 

12 

18 4 

110.6 

202.8 

295.2 

387-7 

480.5 

573-6 

667.0 

13 

20.0 

I 12. I 

204.4 

296.7 

389-3 

482.1 

575-1 

668.6 

14 

21.5 

” 3-7 

205.9 

298.3 

390.8 

483.6 

576.7 

670.1 

15 

23.0 

115-2 

207.5 

299.8 

392.4 

485.2 

578.2 

671.7 

Hi 

24.6 

116.7 

209.O 

3 OI "3 

393-9 

486.7 

579-8 

673 3 

17 

26.1 

118.3 

210.5 

302.9 

395-5 

488.3 

581.3 

674.8 

18 

27.6 

119.8 

212. I 

3 ° 4-4 

397 -o 

489 8 

582.9 

676.4 

19 

29.2 

121.4 

213.6 

306.0 

398.6 

49*-3 

584-4 

677.9 

20 

3°-7 

122 .9 

215.1 

3 ° 7-5 

400. I 

492.9 

586.0 

679.5 

21 

32-3 

124.4 

2 ’ 6.7 

309.1 

401.6 

494-5 

587.6 

681.1 

22 

33-8 

126.0 

218.2 

310.6 

403.2 

496.0 

589.1 

682.6 

23 

35-3 

127.5 

2x9.8 

312.1 

404.7 

497.6 

590-7 

684.2 

24 

36.9 

I29.O 

221.3 

3*3 7 

406.3 

499.1 

592.2 

685.7 

25 

38-4 

130.6 

222.8 

315-2 

407.8 

500.7 

593-8 

687.3 

26 

39-9 

132.1 

224 4 

316.8 

409.4 

502.2 

595-4 

688.9 

27 

41-5 

133.6 

225.9 

3 * 8 -3 

410.9 

503-8 

596-9 

690 4 

28 

43.0 

>35 2 

227.4 

3*9 9 

412.5 

505-3 

598.5 

692.0 

29 

44-5 

130.7 

229.0 

321.4 

414.0 

506.9 

600.0 

693.6 

30 

46.1 

138.3 

230 5 

322.9 

415.5 

508.4 

601.6 

695 * 

31 

47 -6 

139.8 

232.1 

324-5 

4 I 7 - 1 

510.0 

603.1 

696.7 

32 

49 2 

Mi -3 

233.6 

326.0 

418.6 

5”.5 

604.7 

698.2 

33 

5 - 3-7 

142.9 

235 -i 

327.6 

420.2 

513-0 

606.2 

699.8 

34 

52.2 

144.4 

236.7 

329. i 

421.7 

514.6 

607.8 

7014 

35 

53-8 

146.0 

238.2 

330.6 

423.3 

516.2 

609.3 

702.9 

36 

55-3 

H 7-5 

239.8 

332-2 

424.8 

517 7 

610.9 

704.5 

37 

56.8 

149.0 

241-3 

333-7 

426.4 

5 i 9-3 

612.5 

706.1 

38 

58.4 

150.6 

242.8 

335-3 

427.9 

520.8 

614.0 

707.6 

39 

59-9 

152.1 

244.4 

336.8 

429.5 

522.4 

615.6 

709.2 

40 

61.4 

153-6 

245.9 

338.4 

431.0 

523.9 

617.1 

710.7 

41 

63.0 

155-2 

247.5 

339-8 

432.6 

525-5 

618.7 

712.3 

42 

O 5 

*56.7 

249.0 

341-4 

434-1 

527.0 

620.2 

713.9 

43 

66.0 

158.2 

250.5 

343-0 

435-6 

528.6 

621.8 

715.4 

44 

67.6 

159.8 

252 I 

344-5 

437-2 

53 o.i 

623.3 

717.0 

45 

69.1 

"61.3 

253-6 

346.i 

438.7 

531-7 

624.9 

718.6 

46 

70.6 

162.9 

25s.1 

347-6 

440.3 

533-2 

626.4 

720. I 

47 

72.2 

164.4 

256.7 

349-1 

44 1 • 8 

534-8 

628.0 

721.7 

48 

73-7 

165.9 

258.2 

350.7 

443-4 

536.3 

629.6 

723.3 

49 

75-3 

*67 • 5 

259.8 

352.2 

444 9 

537 9 

631.1 

724.8 

50 

76.8 

169.0 

261.3 

353-8 

446.5 

539 4 

632.7 

726.4 

51 

78.3 

170.6 

262.8 

355-3 

448.0 

54 i-o 

634.2 

728.0 

52 

79-9 

172.1 

264 4 

3569 

449 6 

542.5 

635.8 

729 -5 

53 

81.4 

173-6 

265.9 

358-4 

451.1 

544-1 

637.3 

731.1 

54 

82.9 

175-2 

267.5 

360.0 

452.7 

545-1 

638.9 

732.7 

55 

84-5 

176.7 

269.0 

361.5 

454.2 

547-2 

640.4 

734-2 

56 

86.0 

178.2 

270.5 

363-0 

455-8 

548.7 

642.0 

735-8 

57 

87-5 

179.8 

272 . T 

364.6 

457-3 

550.3 

643.6 

737-4 

58 

89.1 

181.3 

273.6 

366 1 

458 9 

55i 8 

645.1 

738.9 

59 

90.6 

182.9 

275.2 

367-7 

460.4 

553 4 

646.7 

74°-5 

60 

92.2 

184.4 

276.7 

369 2 

461.9 

555-0 

648.3 

742.0 













































VER TIC A L A NG U LA T 10 IT IN SKETCHING 



ANGLES OF ELEVATION OR DEPRESSION. 


v y y>- U yv •’*- ** ■ 1 

-» - • <_ ' ^ ‘ 


D — distance in miles, a = angle of elevation or depression; 
k } .=. 5280 ft. X tan a; /z 3 — correction for curvature and re¬ 
fraction. Argument for h 1 is a; argument for h 2 is D. 


Corrections for curvature 
and refraction (always to 
be added algebraically). 


8° 

h , 

0° 

/'t 

10° 

11° 

*1 

12° 

/‘1 

13° 

14° 

15° 

/<! 

/> 


I> 

h. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

miles 

feet. 

miles 

feet. 

742.0 

836.3 

93 1 -o 

1026.3 

1122.3 

1219.0 

1316.5 

1414.8 

1 .O 

0.6 

5-5 

i 7-3 

743 6 

837.8 

932.6 

IO27.9 

1123.9 

1220.6 

00 

w 

m 

1416.4 

I . I 

0.7 

5-6 

18.0 

745-2 

839-4 

934 2 

1029.5 

1125-5 

1222.2 

1319-7 

1418.0 

I 2 

0.8 

5-7 

18.6 

740.7 

841.0 

935-8 

1031.1 

I127.I 

1223.8 

1321.3 

1419.7 

i -3 

1.0 

5-8 

i 9-3 

7483 

842 6 

937-3 

1032.7 

1128.7 

1225.5 

1323.0 

142 r 3 

1 -4 

1 .i 

5-9 

20.0 

749-9 

844. T 

938-9 

io 34-3 

1130.3 

1227.I 

1324.6 

1423.0 

i -5 

i -3 

6.0 

20.6 

75 1 • 4 

845-7 

940-5 

1035.9 

1131-9 

1228.7 

1326.2 

1424.6 

1.6 

i -5 

6.1 

21.3 

753 -o 

847-3 

942.1 

1037.5 

11 . 33-5 

1230.3 

1327-9 

1423-9 

i -7 

i -7 

6.2 

22.0 

754-6 

848.9 

943-6 

1039.1 

1135 2 

1231.9 

1329-5 

1427.9 

1.8 

i -9 

6-3 

22.8 

756 .1 

850.4 

945-3 

IO4O.7 

1136.8 

1233.6 

1331-1 

1429.6 

1.9 

2 . I 

6.4 

23-5 

757-7 

852.0 

946.8 

1042.3 

1138.4 

1235-2 

1332.8 

1431.2 

2.0 

2-3 

6-5 

24.2 

759-3 

853.6 

948.4 

1043.8 

I140.0 

1236.8 

1334-4 

1432.9 

2. I 

2.5 

6.6 

25.0 

760.9 

8.55 • 2 

950.0 

1045.4 

1141.6 

1238.4 

i 335 -o 

1434-5 

2.2 

2.8- 

6.7 

25-7 

762.4 

856.8 

951.6 

IO47.O 

1143-2 

I24O.O 

1337-7 

1436.2 

2-3 

3 -o 

6.8 

26.5 

765.0 

858.3 

953-2 

1048.6 

1144.8 

12417 

1339-3 

M 37.8 

2.4 

3 3 

6 9 

27.3 

765.6 

859-9 

954-7 

1050.2 

1146.4 

12433 

1340.9 

1439-5 

2-5 

3-6 

7.0 

28.1 

767.1 

861.5 

956.3 

1051.8 

1148.0 

1244.9 

1342.6 

1441.1 

2.6 

3 9 

7 -i 

28.9 

768.7 

863.0 

957-9 

1053.4 

1149.6 

1246.5 

1344.2 

1442.8 

2.7 

4.2 

7.2 

29.7 

770.3 

864.6 

959-5 

1055.0 

1151-2 

1248.1 

1345-8 

1444.4 

2.8 

4-5 

7 3 

30.5 

771.8 

866.2 

961.1 

1056.6 

1152.8 

1249.8 

1347-5 

1446.1 

2.9 

4-8 

7-4 

31-4 

773-4 

867.8 

962 7 

1058.2 

1154-4 

1251-4 

1349-1 

1447-7 

3 -o 

5-2 

7-5 

32.2 

775 -o 

869.4 

964-3 

1059.8 

1156.1 

1253.0 

1350.8 

1449.4 

3-1 

5-5 

7.6 

33 -i 

7765 

870.9 

965.9 

1061.4 

H 57-7 

1254 6 

1352.4 

i 45 i-o 

3-2 

5-9 

7-7 

34 ° 

778 1 

872.5 

967-5 

1063.0 

H 59-3 

1256.2 

i 354 -o 

1452.7 

3-3 

6.2 

7-8 

34-9 

779.7 

874.1 

969.0 

1064.6 

1160.9 

1257-9 

1355-7 

1454-4 

3-4 

6 6 

7-9 

35 8 

781.3 

875-7 

970.6 

1066.2 

1162.5 

12595 

1357-3 

1456.0 

3 5 

7.0 

8.0 

367 

782.8 

877-3 

972.2 

1067.8 

1164.1 

1261.1 

1358.9 

1457-7 

36 

7-4 

8.1 

37-6 

784.4 

878.8 

973-8 

1069.4 

1165.7 

1262.7 

1360.6 

1459-3 

3-7 

7-8 

8.2 

38.6 

786 0 

880.4 

975-4 

1071.O 

1167.3 

1264.4 

1362.2 

1461.0 

3-8 

8-3 

8-3 

39-5 

787-5 

882.0 

977-0 

1072.6 

1168.9 

1266.0 

1363 9 

1462.6 

3-9 

87 

8.4 

40.5 

789.1 

883.6 

978.6 

1074.2 

1170.6 

1267.6 

1365-5 

14643 

4.0 

9.2 

8-5 

4 i 4 

790.7 

885.1 

980.1 

1075.8 

1172.2 

1269.3 

1367.1 

1465.9 

4.1 

9.6 

8.6 

42.4 

792.2 

886.7 

981.7 

1077.4 

1173.8 

I27O.9 

1368.8 

1467.6 

4.2 

IO. I 

8.7 

43-4 

793 8 

888.3 

983.3 

1079.0 

1175-4 

1272.5 

1370.4 

1469.2 

4-3 

10.6 

8.8 

44-4 

795-4 

889.9 

984.9 

1080.6 

1177 .O 

1274-1 

1372.1 

1470.9 

4.4 

II . I 

8.9 

45-4 

797.0 

891.5 

986 5 

1082.2 

1178.6 

1275-7 

1373-7 

1472.5 

4-5 

11.6 

9.0 

46.4 

798.5 

893.0 

988.1 

1 1083.0 

1180.2 

1277.4 

1375-3 

1474.2 

4.6 

12 . I 

9 -i 

47-5 

800.1 

894.6 

989.7 

1085.4 

1181.8 

1279.0 

1377.0 

1475-9 

4-7 

12.7 

9.2 

48.5 

801.7 

896.2 

99 i -3 

1087.0 

1183.4 

1280.6 

1378.6 

1477-5 

4.8 

13.2 

9-3 

49-6 

803.2 

897.8 

992.9 

1088.6 

1185.0 

1282.2 

1380.3 

1479-2 

4.9 

13-8 

9-4 

50-7 

804.8 

899.4 

994-5 

I090.2 

1186.7 

1283.9 

1381.9 

1480.8 

5 -o 

14-3 

9-5 

5 i -7 

806 4 

900.9 

996.0 

1091.8 

1188.3 

1285.5 

13835 

1482.5 

5 -i 

14-9 

9.6 

52.8 

807.9 

902.5 

997.6 

1093.4 

1189.9 

1287.1 

1385 2 

1484.1 

5-2 

15-5 

9-7 

53-9 

8-9.5 

904. T 

999-2 

1095.0 

ii 9 i -5 

1288 8 

1386 8 

1485.8 

5-3 

16.1 

9.8 

55 -i 

811.1 

905 -7 

1000.8 

1096.6 

ii 93 -i 

1290.4 

1388.5 

1487.5 

5-4 

16.7 

9-9 

56.2 

812.7 

907.3 

1002.4 

1098.2 

1194.7 

1292.0 

1390.1 

1489.1 

5-5 

i 7-3 

io.o 

57-3 

814.2 

908.8 

1004 O 

1099.8 

1196.3 

1293-7 

1391.8 

1490.8 





815.8 

910.4 

1005.6 

1101 4 

1197.9 

1295-3 

1393-4 

1492.4 





817.4 

912.0 

1007.2 

1103 0 

1199.6 

1296.9 

1 . 395-0 

I 494 -I 





819.0 

913 6 

1008.8 

1104.6 

1201.2 

1298.5 

1396-7 

1495.8 




1 

820.5 

9 t 5.2 

IOIO.4 

i106.3 

1202.8 

I3OO.2 

1398.3 

1497.4 





822.1 

916.7 

1012 .O 

1107.9 

I204.4 

1301.8 

1400.0 

1409.1 





823.7 

918.3 

1013.6 

1109.5 

1206 0 

1303-4 

1401.6 

1500.7 





825.2 

919.9 

1015.2 

1111.1 

I207.7 

1305.0 

'4033 

1502.4 





826 8 

921.5 

1016.8 

IT 12.7 

1209 3 

1306 7 

1404.9 

1504.1 





828.4 

923.1 

1018.4 

H14.3 

1210.9 

1308.3 

1406.5 

15057 





830.0 

924.7 

1020.0 

1115-9 

1212.5 

1309.9 

1408.2 

1507-4 





831-5 

926.2 

1021.5 

1117-5 

1214.I 

1311.6 

1409.8 

I5O9 O 





8331 

927 8 

1023.1 

II19.I 

1215.8 

1313-2 

1410.5 

1510 7 





834.7 

929-4 

1024 7 

I120.7 

1217.4 

1314.8 

1413-1 

1512.4 





836.3 

93 1 -o 

1026.3 

T122.3 

1219.0 

1316.5 

1414.8 

1514.0 



















































366 


TRIG ONOME TRIC LE VEL ING. 


manner as vertical angulation is conducted in the course of 
traverse-work (Art. 163). While sketching, the topographer 
has before him on his plane-table board all of the plotted 
control, including positions of triangulation stations, of 
adjusted traverse lines, and of points intersected from the 
traverses (Art. 84). Assuming now that he has been sketch¬ 
ing for some little time by means of an aneroid adjusted at 
some fixed elevation along the route of his traverse (Art. 176), 
and it becomes desirable either to check the aneroid or to deter¬ 
mine the elevation of some nearby point which he is sketching. 

Setting up the plane-table at a known position he reads an 
angle with the telescopic alidade or vertical-angle sight-alidade 
(Arts. 59 and 62) to some house on a neighboring road or 
hillside, or to some near-by summit which is plotted on the 
map, and this angle, with the distance measured on the plane- 
table sheet, furnishes the data from which to compute his 
height (Art. 161). Or, vice versa, knowing his elevation by 
an aneroid which has been recently checked, or being at some 
point the height of which has been determined by spirit-level 
or previous vertical angulation, he may determine the eleva¬ 
tion of other located points which are within view, as a house 
on a neighboring road or hillside, or a summit, by reading an 
angle to them with the alidade and measuring the plotted 
distance on the plane-table. In this way he may keep eleva¬ 
tions placed ahead of him on adjacent roads or hills over 
which he expects to travel, or he may bring those elevations 
to him after he has reached such positions. 

In all such vertical angulation, either performed in the 
course of traverse-work or of sketching, the topographer must 
bear in mind clearly the fact that the accuracy of the determi¬ 
nation is dependent on the distance and on the difference of 
elevation or degree of the angle read. The smaller the dis¬ 
tance the steeper may be the angle, and yet produce no great 
error; the greater the distance the smaller must be the angle. 
(Verify by Table XV.) Reliance should not be placed 


VERTICAL ANGULATION FROM TRAVERSE. 367 


where the scale is about one mile to one inch and the con¬ 
tour interval about 20 feet on angles exceeding 2 degrees at 
distances of 2 or 3 miles. The same proportion holds true 
for different contour intervals and scales, and the degree of 
accuracy with which the base elevation is determined and the 
platted positions are fixed. 

163. Vertical Angulation from Traverse. —In traversing 
with the plane-table opportunity frequently arises to obtain 
the elevation of near-by points as referred to the known heights 
of the traverse stations; or, vice versa , the heights of the 
traverse stations may be obtained by vertical angulation from 
points the elevation of which is already known. This is 
done by reading angles with the telescopic alidade or with 
the vertical-angle sight-alidade (Arts. 59 and 62) from some 
traverse station to the object the difference in elevation 
of which is to be determined. The angle read, together 
with the distance measured on the plane-table board, furnish 
data from which to compute or to obtain from tables the 
differences in elevation. By this means the heights of trav¬ 
erse stations may be frequently obtained and the aneroid 
checked thereby, and then the heights of minor surrounding 
points may be obtained at intermediate stations on the trav¬ 
erse from the adjusted aneroid elevations. 

An example of traverse notes accompanied by vertical 
angulation is as follows: 


Date , Nov. 16, 1898. Traverse from Jonesboro, N. C.,to Walnut Grove, N. C. 


Remarks. 

Station. 

Aneroid. 

Cor. Curv. 
and Refr. 

Height of 
Signal. 

Point. 

Level. 

! Angle. 

Dist.,Miles. 

Diff. Eleva¬ 

tion. 

Elevation. 

House.. 

25.1 


+ 2' 

o' 

15 00 

i 5 ° i 5 ' 

- o° 13' 

1.86 

- 38 ' 

3069' 

3°33 

Flag ... 

— 26.25 


+ 1 

O 

14 56 

»4 50 

4- 0 06 

1.46 

+ 15 

3049 

Flag ... 

— 27.26 



O 

15 00 

M 53 

+ 0 C7 

.69 

4 - 7 

3056 


The angle used in the computation is the difference be¬ 
tween the angle read when pointing at the station sighted 

























3 68 


TRIGONOMETRIC LEVELING. 


and that when the telescope is horizontal, as shown by the 
striding-level (Art. 161). The distance is measured directly 
by stadia, odometer, chain, or upon the plane-table sheet 
(Arts. 102, 98, and 99). The difference of elevation is ob¬ 
tained by computation (Arts. 161 and 164). The correction 
to curvature and refraction (Art. 166) is applied to the dif¬ 
ference of elevation and gives a resulting elevation in the 
last column. Account must be taken of the height of in¬ 
strument, about 4^ feet. Such an example would apply 
either to observations taken from stations to points about the 
traverse or, as in this case, to backsights and foresights on 
the traverse line. The fact of the sight being a backsight or 
a foresight is indicated by a + or — sign in the index 
column, which affects the application of the correction for 
curvature and refraction, as the latter is always algebraically 
positive. 

164. Trigonometric Leveling, Computation. —To de¬ 
termine the difference of elevation by zenith distances, let 

Z and Z' — the measured zenith distances at the two sta¬ 


tions ; 


D = the distance between stations, in meters; 

R — radius of curvature of the arc joining the two, 
in meters; 

C = angle at the center of the earth subtended by 
this arc; and 


H and H' = the heights of the two stations observed;—then 


^ T~) // • 

R sin 1 


(28) 


and 



D sin i(Z — Z') 


(29) 


cosin \{Z — Z' -f- C) 





TRIGONOMETRIC LEVELING , COMPUTATION . 369 


The value of A? or of ~ may be computed for 

different latitudes and for varying angles from Table XVI, 
based on Clarke’s Constants and taken from the report of the 
U. S. Coast and Geodetic Survey for 1877. 

Table XVI. 

LOGARITHMS OF RADIUS OF CURVATURE, R, IN METERS. 



Azimuth. 

Latitude. 


26° 

28° 

30° 

32° 

34° 

36° 

Meridian. 

O 

O 

6-802479 

6.802597 

6.802722 

6.802852 

6 802988 

6.803129 

6.803274 


5 

2498 

2615 

2739 

2869 

3004 

3M5 

3289 


IO 

C553 

2669 

2791 

2919 

3°5 a 

3190 

3332 


*5 

2644 

2756 

2875 

3000 

‘3*3° 

3265 

34°4 


20 

6766 

2875 

2990 

3111 

3236 

3366 

3500 


30 

3°93 

3192 

3296 

3405 

35i8 

3636 

3757 


40 

3496 

358o 

367 1 

3766 

3864 

3967 

4072 


50 

3923 

3994 

4070 

4150 

4233 

43*9 

4407 


60 

4325 

4384 

4446 

4512 

4580 

4650 

4723 


70 

4653 

4702 

4753 

4807 

4863 

4921 

4980 


75 

4776 

4822 

4869 

49*8 

4969 

5022 

5076 


80 

4867 

4909 

4953 

4999 

5047 

5097 

5*48 


85 

4923 

4963 

5006 

5049 

5096 

5*4-3 

5*92 

Perpendicular ... 

90 

6.804942 

6.804981 

6.805023 

6.805066 

6.805112 

6.805159 

6.805207 



O 

1 00 
m 

4°° 

42° 

44° 

46° 

48° 

50° 

Meridian. 

O 

6.803422 

6.803573 

6.803726 

6.803880 

6.804035 

6.804189 

6.804342 


5 

343 6 

3586 

3739 

3892 

4045 

4199 

435* 


IO 

3478 

3626 

3775 

3926 

4077 

4228 

4378 


15 

3546 

3690 

3835 

3982 

4130 

4277 

4423 


20 

3637 

377 6 

3917 

4059 

4201 

4343 

4484 


3° 

3880 

4006 

4133 

4262 

439i 

45*9 

4647 


40 

4179 

4289 

4400 

45H 

4623 

4735 

4846 


50 

4498 

4590 

4683 

4777 

4871 

4965 

5058 


60 

4797 

4873 

4949 

5025 

5 ,Q 4 

5181 

5257 


70 

5041 

5 io 4 

5166 

5229 

5293 

5357 

5420 


75 

5133 

5 I 9° 

5248 

5307 

5364 

5423 

548* 


80 

5201 

52.54 

53o8 

5.36.3 

54*7 

5472 

5526 


85 

5242 

5294 

5345 

5397 

5450 

5502 

5554 

Perpendicular... 

90 

6.805256 ,6.805307 

6.805358 

6.805409 

6.805460 

6.805512 

6.805563 


Example. 

K — 2393 i m .6.distance between two stations, Santa Cruz and Mount 

Bache, California. 

Z — 87° 35' 01".06 observed at Santa Cruz station, reduced to ground at 

Mount Bache. 

Z' — 92 0 35'34".20 observed at Mount Bache, reduced to ground at Santa 


Cruz station. 

L — 37 0 02'.mean latitude of the two stations. 

Angle = 51 0 55'.angle made by line with the meridian. 

























































370 


TRIG ONOME TRIC LE VELING. 


Computation of h — h'. 


log K. . 4.3790 

colog R sin 1" 8.5101 


log C . 2.8891 

c= 774.56 


Z' - Z .... 5 00 33.14 
\{Z'~ Z ).. 2 30 16.57 
z'— z 4- c 5 13 28.06 
— Z-j-C) 2 36 44.03 


log A".. 4.3790 

log sin \{Z’ — Z)... . 8.6405 
colog cos \{Z' — Z- j- C) o. 0004 


3- OI 99 


m 

Difference in height. 1046.90 

Santa Cruz station above mean tide—by spirit-level. 108.87 


Mount Bache above mean tide 


1155-77 


165. Errors in Vertical Triangulation. —In this class of 
leveling there are several sources of error, the most import¬ 
ant of which, perhaps, is the refraction of the atmosphere. 
In vertical angulation (Art. 161) this may be compensated 
by applying approximate or mean corrections. In more pre¬ 
cise trigonometric leveling the amount of refraction should be 
determined by direct observation in order that the correction 
may be most accurately applied. The correction of largest 
amount is that for curvature , but this is accurately known. 
Other sources of error are due to— 

1. Errors of measurement of the distance between the 
objects; 

2. Errors of the instrument, both of graduation and of 
level-bubble; and 

3. Errors of pointing on the signal, or its height or defini¬ 
tion. 

Most of the errors of instrument excepting those of gradu¬ 
ation may be eliminated by taking direct instrumental obser¬ 
vations on the object sighted and reading the level and verti¬ 
cal circle, then reversing the instrument in its wyes and again 
reading the angle. Half the difference of the reading would 
thus be corrected for the difference of level. Shifting the 
vertical circle and repeating the reading would aid slightly in 
further reducing the errors of graduation and observation. 
These errors are small, however, compared with the errors 













REFRACTION AND CURVATURE. 


371 


arising from refraction, which can only be partially eliminated 
by observing on different days in order to get different atmos¬ 
pheric conditions. The best results in trigonometric leveling 
are to be obtained at such times of the day as refraction is 
least. 

166. Refraction and Curvature. —The coefficient of re¬ 
fraction or the proportion of intercepted arc is determined 
from the observed zenith distances to two stations, the rela¬ 
tive altitudes of which have been determined by the spirit- 
level ; or from reciprocal zenith distances, simultaneous or 
not, under the assumption that the mean of a number of ob¬ 
servations taken under favorable conditions will eliminate the 
differences of refraction found to exist even at the same mo¬ 
ment at two stations a few miles apart. The difference of 
height from trigonometric leveling being dependent upon the 
coefficient of refraction multiplied by the square of the dis¬ 
tance, it is therefore evident that the longer the line the 
greater will be the error caused by any uncertainty in the co¬ 
efficient, and that there is therefore a limit to the distances 
for which any assumed mean values of refraction can be de¬ 
pended upon for accurate results. 

The coefficient of refraction is the angle of refraction di¬ 
vided by the arc of the earth’s circumference intercepted 
between the observer and the station observed. 

Let c = angle at the earth’s center, subtended by two 
stations, j and ; 
f — angle of refraction ; and 
r— coefficient of refraction;—then 


f= C --\{Z’ + Z- .80°), 


/ 

c 


and r=- .(30) 

The value of c in seconds can be found from the expression 

d 


c n = 


y sin I 


tn 


(31) 




372 


TRIG ONOME TRIC LE VELING. 


in which d is the distance between the two stations, and y is 
the radius of the earth. 

Refraction is least and is comparatively stationary between 
9 A.M. and 3 P.M. It is greatest early in the morning, and 
after 3 P.M. it increases in amount and variation to a maxi¬ 
mum during the night. The value of the coefficient of re¬ 
fraction r differs according to various observations from 0.06, 
observed by the U. S. Lake Survey in central Illinois, to 0.08, 
observed by the U. S. Coast and Geodetic Survey in New 
England near the sea-level, and in the interior of the coun¬ 
try or at considerable altitudes between 0.065 and 0.07. 

The amount and method of application of the correction 
for the curvature of the earth have been briefly indicated in 
Articles 160 and 161. The amount of this correction for 
various distances is more fully shown in Article 239, which 
gives also in tabular form (Table XXXI) the amount of re¬ 
fraction and the combined amount of the two. 

167. Leveling with Gradienter. —The gradienter screw 
may be used as an adjunct to a tachymetric instrument, 1st, for 
the purpose of measuring vertical angles and thus determining 
differences of elevation; and, 2d, as a telemeter for the meas¬ 
urement of horizontal distances (Art. 114). The gradienter is 
a tangent screw with micrometer head attached to the horizon¬ 
tal axis of the telescope. Originally, as its name implies, the 
gradienter was employed in locating grades on railway and 
canal surveys. It has also been satisfactorily employed by 
the writer in interpolating contours on uniform slopes espe¬ 
cially in the survey of reservoir sites. 

To locate a grade of 2 \ per cent, for example, which is a 
grade of 2 \ feet per hundred, the telescope is leveled and the 
head of the gradienter screw read. Then, for a screw gradu¬ 
ated so that one revolution corresponds to one foot in 100, 
the same must be revolved 2^ turns, when the line of sight 
of the telescope will be on the grade desired. The gradienter 
may be employed in measuring elevations b}' means of verti- 


LEVELING WITH GRADIENTER. 


373 


cal angles in terms of the tangent. For, with a knowledge of 
the horizontal distance obtained by the gradienter (Art. 114) 
or otherwise, a small vertical angle may be read by the 
micrometer screw, or large ones read with the vertical arc of 
the instrument supplemented by the micrometer screw, and 
this vertical angle in connection with the distance gives the 
data from which to compute the difference of height (Art. 
161). 


CHAPTER XVIII. 


BAROMETRIC LEVELING. 

168. Barometric Leveling'. —Barometric leveling is espe¬ 
cially adapted to finding the difference between two points at 
considerable horizontal or vertical distances apart and which 
are unconnected by any system of plane survey. As a result 
it is the most speedy though least accurate of the methods of 
leveling. It is, however, very useful in making exploratory 
or geographic surveys over extensive areas or for making re¬ 
connaissance surveys for railroads or similar engineering works. 
Barometric hypsometry is frequently the only means by which 
approximate elevations may be determined in the progress of 
rough or reconnaissance surveys. 

Two general classes of instruments are employed in the 
making of hypsometric observations in such surveys, namely: 

1. The cistern or mercurial barometer; and 

2. The aneroid. 

Both of these instruments are dependent upon the differ¬ 
ences of atmospheric pressure at two different elevations. The 
higher we rise above sea level the less the depth of atmos¬ 
phere above us, and consequently the less its weight and the 
height to which it will raise or counterbalance a column of 
mercury. Thus, if the barometer records 30 inches of pres¬ 
sure, that is, sustains a column of mercury thirty inches in 
height, at the level of the sea, it will, at an elevation of 1000 
feet, sustain a column of approximately 28.9 inches. The 

374 


METHODS AND ACCURACY. 


375 


aneroid is a much more compact instrument than the mercu¬ 
rial barometer, more portable, and is carried in a metal case 
similar to that of a large watch. 

169. Methods and Accuracy of Barometric Leveling.— 
The differences of atmospheric pressure as recorded by barom¬ 
eters is affected by the temperature, and compensation for 
temperature must be made in order to obtain the best results 
from barometric measurements. Ordinarily the aneroid or 
mercurial barometer is adjusted at the elevation of the start¬ 
ing-point, and readings are taken at various points the heights 
of which are to be determined and the elevations' to which 
they correspond are computed therefrom. More accurate re¬ 
sults can be obtained by the synchronous readings of two 
barometers, one of which remains stationary at a known ele¬ 
vation, while the other is read at points the heights of which 
are to be determined, and the difference between the two 
gives the data from which to compute the differences in 
height. 

As the weight of the atmosphere and the consequent record 
of the barometer are affected by humidity far more than by 
temperature, the readings of two instruments which are af¬ 
fected by approximately the same atmospheric conditions 
give a better relative difference in height than could be ob¬ 
tained by the reading of one. Forty or fifty determinations 
of elevations by mercurial barometer were obtained ten or 
fifteen years ago in widely separated regions in the course of 
the early hyposometric surveys of the U. S. Geological Sur¬ 
vey at points the elevations of which were known from spirit¬ 
leveling. It is interesting to note that the average error in 
these determinations was but a little over 8 feet, and the ex¬ 
treme error 17 feet. It is thus seen that under the most 
varying conditions where a barometer is carefully and well 
used fairly satisfactory results may be looked for, though un¬ 
accountable atmospheric disturbances may give results in 
error over 100 feet under apparently favorable conditions. 


376 


BA ROME TRIG LE VELING. 


170. Mercurial Barometer. —The mercurial barometer 
consists of two parts, the cistern and the tube. The cistern 
is made up of a glass cylinder, E , through which the surface of 
the mercury can be seen ; an upper inclosing plate, G , through 



Fig. 113.—Section through Ctstern and Tube of Mercurial 

Barometer. 


which the lower end of the barometer tube, t , passes and to which 
it is fastened by a piece of kid leather, so as to make a strong 
but flexible joint. (Fig. 113.) Below these and to this plate 
is attached by long screws, P, a lower metal cup from which is 
suspended a wooden reservou or cistern, M, the bottom portion 
of which is formed by a kid orchamois-leather bag, N. This is 













































































































































































MERCURIA L BA ROME TER. 


177 


so contrived that it may be raised or lowered by means of an 
adjusting-screw, 0 , and the surface of the mercury, as seen 
through the glass cylinder, can be brought to exact contact 
with an ivory pointer, q , when the instrument is to be read. 
When being transported the adjusting-screw is turned up 
tightly until the mercury completely fills the tube, when the 
latter can be inverted and carried with the cistern end upper¬ 
most, so as not to be liable to breakage by the jar or shock of 
the mercury splashing against the upper end of the glass tube. 
When being read, and after the index-point has been brought 
to exact contact with the mercury surface, a sliding scale, on 
which is a vernier, is brought to corresponding contact with 
the upper surface of the mercury in the tube by turning a screw, 
D, and the reading on the vernier is recorded. 

Though the barometer may be received filled from the 
maker, one who uses it should understand how to fill it in 
case of the not improbable breakage of the tube. The mer¬ 
cury used in filling a barometer should be mechanically pure, 
and is best transported in short iron tubes made of sections of 
gas-pipe. It is rarely necessary to boil the mercury in the 
tube to expel moisture or air, as is the general practice, since 
barometers can with a little practice be filled in a sufficiently 
satisfactory manner with cold mercury. 

If a new glass tube is to be inserted, this should be trans¬ 
ported in a box for safe packing, and the ends should be sealed 
until required for insertion, when the lower end is to be cut 
off with a sharp file and the edges filed straight and smooth, 
or, better, heated over a flame until they are rounded by 
fusion. The mercury should then be dropped in the open end 
of the tube slowly through a clean, rough paper funnel with a 
hole so small as only to let the mercury through a drop at a 
time, thus filtering it. When the tube is filled within a quarter 
of an inch of the top, the open end must be closed with a piece 
of chamois placed over the thumb, and the bubble of air which 
remains is to be run back and forth in the tube by inclining it 


37S 


BAROME TRIC LE VELING 


so as to gather together all small air-bubbles adhering to the 
inside of the glass. 

When all the bubbles have been collected turn the tube 
up again so that the large bubble shall pass to the open end. 
This should then be completely filled with mercury, and a 
little of the mercury may be again let out ^ id the same oper¬ 
ation repeated with an expanded or vacuum air-bubble until 
all the air has been removed. This can be distinguished by 
letting < the column of mercury run sharply against the closed 
end of the tube, when it will give a clear metallic click if 
there is no free air in the tube. The tube is then placed open 
end upward, again filled to overflowing with mercury, top 
plate and glass plate on upper half of wooden cistern screwed 
tightly on. The cistern is then filled with mercury to over¬ 
flowing, and the lower half, carrying the kid bag is placed on 
it and the two halves of the cistern joined together. Now 
screwing on the outer metal case and having the adjusting-screw 
tightly fastened up, the instrument may be reversed or placed 
in its upright position, when it is ready for use. A similar 
operation has to be repeated in case the mercury in the cistern 
has become dirty or the ivory point dirty from oxidation, it 
being necessary to first tighten up the adjusting-screw, invert 
the instrument, and remove the lower half of the cistern. 

271. Barometric Notes and Computation. —There are 
several modes of keeping barometric notes as well as of 
computing them, according to the formulae employed. The 
general theory on which barometric work is computed depends 
upon the fact that at sea-level the weight of the column of 
atmosphere above any given point is approximately 1 5 pounds 
to the square inch, which is sufficient to raise a column of 
mercury in a vacuum tube to the height of 30 inches. As 
one ascends the pressure diminishes because of the diminution 
in the height of the column of air above. But this diminu¬ 
tion is not in a simple ratio depending on altitude because 
there are varying densities in the strata of air produced largely 


BAROMETRIC NOTES AND COMPUTATION. 


379 


by retained moisture and wind-pressure. Moreover, each 
succeeding layer of air is less dense than that which underlies 
it by the weight of the stratum beneath it. The difference 
in heights of any two places is equal to the difference between 
the logarithms of the air-pressures at those two places mul¬ 
tiplied by a certain constant distance. It is this relation 
which gives the first and principal term in the various tables 
for reducing barometric work. Numerous determinations of 
the pressure constants have been made, and these produce 
the principal differences in the various tables. 

The more important barometric tables are dependent origi¬ 
nally on Laplace’s formula and the use of his coefficients. 
One of the tables first and most extensively used in this 
country is known as Williamson’s Table, having been first 
expounded in a treatise by Lieutenant-Colonel Williamson on 
the ** Use of the Barometer.” The tables generally accepted 
now as giving the best results are A. Guyot’s. 

Laplace s formula reduced to English measures is as 
follows: 


Z — log 60158.6 Eng. ft. 

Jrl 


0 + 


t t — 64) 


9OO 

(1 -j- 0.0026 COS 2 L) 
Z + 52252 


( x + 


+ 


h 


) 


20886860 10443430)' 


in which 


h — the observed height of the barometer, 
r — the temperature of the barometer, 
t — the temperature of the air, 
h' — the observed height of the barometer, 
t' — the temperature of the barometer, 
t' — the temperature of the air, 

Z = the difference of level between the two barometers; 
L — the mean latitude between the two stations; 


at the lower 
station ; 

at the upper 
station. 







3 80 BA ROME TRIG LE VELING. 

H — the height of the barometer at the upper station reduced 
to the temperature of the barometer at the lower 
station, or 

H—h' {i + o.oooo8967(r — r') j-. 

The expansion of the mercurial column for i° Fahrenheit 
= 0.00008967; 

The increase of gravity from the equator to the poles = 
0.00520048 or 0.0026 to the 45th degree of latitude; 

The earth’s mean radius = 20,886,860 Eng. ft. 

An extremely interesting method of computing differ¬ 
ences of elevations barometrically was devised by Mr. G. K. 
Gilbert of the U. S. Geological Survey. Mr. Gilbert made 
an entirely new departure in barometric measuring. He 
abandoned Laplace’s formula, substituting a new formula 
involving none of his constants and having but a single ele¬ 
ment in common. The old method, that based on Laplace’s, 
and by which Guyot’s and Williamson's Tables were prepared, 
was dependent on the thermometer and the difference of 
temperature as recorded by it. The new method abandons 
the thermometer and employs the barometer-alone. 

Gilbert decided that there was an atmospheric gradient; 
that is, that the difference of atmospheric pressure between 
two points at different altitudes differed in some proportion 
to these altitudes. Thus a plane passing through the sum¬ 
mits of verticals erected above the two points is inclined in 
some direction because the pressures are on unequally differ¬ 
ent altitudes. Gilbert determined that there were diurnal 
and annual variations in this gradient, and that in order to 
properly determine difference of altitude by the barometer 
the gradient must be considered, and his mode of so doing is 
to establish two-base barometer stations, one as high as the 
highest of the points the elevations of which are to be deter¬ 
mined, the other as low as the lowest. These should be read 
synchronously at intervals, say of one hour, and the moving 


EX A MPL E OF BA ROME TRIG COMP U TA 7V0JV. 3 8 I 

barometer is Corrected by reduction, not to one-base barometer 
but to two, so that it can be placed in its gradient somewhere 
between the two barometers which are at known altitudes. 

172. Example of Barometric Computation. —Below is 
given an example of an observation made by a moving barom¬ 
eter at McKenzie Mountain, N. M., while at the same hour 
a station barometer was observed at Fort Wingate, N. M., the 
altitude of which is known. The station or base barometer 
was assumed to be without an instrumental error. The 
. moving barometer was compared with it at the beginning of 
the season, May 1, and was reduced to it by first reducing 
the readings to 32 degrees Fahrenheit and then subtracting 
the readings of the moving from the base barometer. The 
five comparative readings ranged between -f- .002 and — .005, 
with a mean of — .003 inches as the error of the moving 
barometer. "7 


BAROMETRIC COMPUTATION. 

Observations at Fort Wingate, IV. M. 


Date. 

1 

Hour. 

Barom. No. 

Upper Vernier. 

Lower Vernier. 

Att. Ther. 

Temp. Cor. 

Inst. Error. 

Total Cor. 

Reduced Read¬ 
ings. 

Thermom. 

cri 

>. 

u 

Q 

CQ 

4>> 

V 

£ 

Mav *21. 1883 . 

9 A.M. 
IO A.M. 

2606 

2606 

23-512 

23 502 

14.789 
14.780 

75°-5 

79 ° 

- .099 

— .106 

O 

O 

— .OgQ 

— .I06 

23 - 4'3 
23• 39 6 
I 3 - 4°4 

76° 

78 

77 

46 ° 

44 

45 

4 4 4 4 44 

Means..... . 






. 





Observatio'ns at McKenzie Mountain, IV. M. 


Mav q 1 „ 1882. 

9 A.M. 
IO A.M. 

2679 

2679* 

30.823 

30-819 

22.I24 
22.119 

58 

6l 

- -058 

— .064 

- .003 

- -003 

— .061 

22.063 

22.052 

22.057 

c6 

40 

41 

40. < 

44 44 44 

— .067 

59 

S7. ^ 

Means . 













The computation by the Guyot method is illustrated in 
the following example side by side with a computation by 
the Williamson method in order that the difference between 
the two may be noted. The terms and D t {H) are 

obtained from Table XVII, the argument for D f (h) being 
the height of the barometer at the base station, and the 


















































jS2 


BAROMETRIC LEVELING 


BAROMETRIC DETERMINATION OF HEIGHTS. 

FIELD SEASON, 1883. 

Party No. 1. Division of Fort Wingate, N. M. 

H. M. Wilson , Computer . 
Observations recorded in books No. 306 and No. 309. 


Names of Tables, etc. 

Williamson’s 

Computation. 

Guyot’s Computation. 

Date.. 

May 31, 9 and io a. 

M. 

No. of synchronous obs. 

2 

2 

Lower station... 

Wingate 

Wingate 

Upper station. 

McKenzie 

McKenzie 

Bar. at 32 0 \ „ 

23.404 

23.404 


22.057 

22.057 

fu 

77 

77 

Temperature \ 

57-5 

57-5 

[t + e - . 

134-5 

134-5 

( a —. 

.130 


Humidity •< a — . 

.284 


(a -f a! — . 

.414 


Latitude =. 

35 ° 30 ' 

35 ° 30' 

L>i(k) 

22299 

22216 

Di(H) 

20745 

20667 

1st approx. =.... 

1554 

1549 

Da =. 

112 

122 

2d approx. .. 

1666 

1671 

Dm .. 

2 

2 

Div -. 

4 

4 

Dy - . 

1 

I 

3d approx. .. 

1673 

1678 

Dv i =. 

22' 


Dy ii =. 

io'^ 


Correct for (a + a') = . 

4 


Diff. of altitude =. 

1677 

1678 

Altitude of reference station =. 

6978 

6978 

Altitude of new station, feet = 

8655 

8656 

Remarks: 













































GUYOT'S BAROMETRIC TABLES. 


383 


argument for D{H) the height of the moving barometer. If 
the new station be lower than the base, the difference between 
D\Ji) and D t (H) is given a negative sign. The corrections 
D u , D //y , etc., are added to the first approximate result regard¬ 
less of its signs, attention being paid to the signs of the cor¬ 
rections, which are generally positive. 

The correction D n is the product of the first approxima¬ 
tion into the factor found in Table XVIII, the argument for 
which is t — t’ or the sum of temperatures of the detached 
thermometers of the two stations. When the humidity correc¬ 
tion is used the relative humidities are first found from Lee's 
Tables, the arguments being the difference between the wet 
and dry bulbs and the reading of the wet bulb, though this 
correction scarce affects the result appreciably and may be 
omitted. 


173. Guyot’s Barometric Tables. —Table XVII gives in 
English feet the value of log H or h X 60158.6 for each 
hundredth of an inch from 12 to 31 inches of barometric 
pressure. The additional thousandths are obtained in a sepa¬ 
rate column. 

Table XVIII gives the correction 2.343 feet X (r — r') 
for the different temperatures of the barometers at the two 
stations; and as that-at the upper station is generally lower, 
r —- r' is generally positive and the correction negative. 
This correction becomes positive only when the temperature 
of the upper barometer is higher. 


Table XIX shows the correction D' 


Z + 52252 
20886860 


to be 


applied to the approximate altitude for the decrease of grav¬ 
ity on a vertical acting on the density of the mercurial 
column. It is always added. 

h 

Table XX furnishes the small correction - for 

10443430 

the decrease of gravity on a vertical acting on the density of 
the air. This correction is always added. 




3«4 


BA ROME TRIC L E VELING. 


H 

W 

w 

Ui 

. o 

■" H 

. .■.(/) 

O 

HH 

Q 

< 

. W 

t-H p/ 

M 

> u 

X 5 
w 5 

r! O 

? 3 

ffl 

pH 

o 

£ 

o 

M 

H 

U 

P 

Q 

W 

P 4 


c 

o 

• »—I 

■H* 

c 3 

w 

c /3 

u 

V 

w 

• H 

<U 

4-> 

rt 

i_ 

<u 

■*—» 

<u 

E 

o 

V* , /—N 
tn 

,Q C 

f-B 1 

-G 3 
*-* X 
’**- 'u 

0 C 
4-) O 
X U 
bfl (/) 
3 


X! 


T 3 rt 
OJ 3 ! 
•> w 

F* o 

1- t/> 

V ~ 
<A S 
•O - 
o 


.2 
c 
o 

t w 
t -1 33 


1) 

O 


” a 

c 5/3 

« a 

6 2 

3 -44 

bo ts 

U V 

< § 
h 
■*-» 

• ^ 
<i W 

s- w 

O 


bo 

o 

X 

CO 

1-0 

co 

U~1 

1-4 

O 

O 

II 


P — • C /3 

!u*s 

V- H v» 
c 3 ♦-» td 
CQ 


</3 

x 

4-> . 
-3 
C 
rt’ 
C /3 

3 

O 

X' 

H 


x 

o 

c 

c 

rt 


C /3 

X 

4-1 

•a 

<u 

u 

•a 

c 

3 

DC 


o 

u 

rt ■ 


O w N n't 
N N N N N 


in o co O' 
ci a « ct M 


O h ci co 3 
co co co co co 


mo co O' 
CO CO CO CO CO 


X 


4 -* 



M Cl 

Cl CO 

3 m O 

0 

r^- 




O'CO 0 


<u 



• • 

• • 

• • • 

• 

• 




• 

• • 

n 


<L> 



Cl 3 - 

O CO 

O Cl 3 

0 

00 




Hi 

co m 

c 


Uh 





M W *-( 

1 —< 

t—» 






p 

h* 

o 





w d 

co 3 m O t" 

CO 

O 




Hi 

Cl co 



■h 

O CO 

Cl 

0 6 

3 O 

cm to 

Oco 

Hi 

O rt 

CO O' O co Cl 


os 

u 

CO CO 


O' O' ; 


O Cl 3 

rt 

co 

Hi 

r ^ Hi 

3 in 0 

3 Ci; 


o 

in oo 

O' 0 

H 4 Cl 

Cl CO CO 

co 

co 

CO 

Cl Cl 

- O 

O' 


• 

G 

w 

O' M 

CD 

to CO 

O Cl 

3 0 oo 

O 

CM 

30 co 

O Cl 

cn 

tn 



3 in 

ID 

to to 

O O 

0 cO 0 

i ^ 


rN 


co co 

co 

CO CO 



4-1 

'tH 

O. rt O 

O' M 

O m co 3 3 / 

O 

0 3 

3 O 

HI CO 

HI 

O' in 


X 

bi 

ci 

00 co 

0 coco ci 3 

3 CO 

H 

r ^ ci 

in O 


tn cd 


c 

co m o 

00 

O' 0 

0 C 4 h 4 

M 

H -1 

Ht 

O O 

e'en 

tNiO m 


• 

W 

0 s M 

co 

to 

O' Cl 

30 CO 

0 

Cl 

30 co 

O'' Hi 

CD 

tn 



3 IT) 

IT) 

LO 10 

to VC 

vC O O 





f'- CO 

CO 

co co 



4-1 

3 O' co Cl 

co co 

COH CO 

O 

M 

co 

0 0 

OO fN 

Hi . 

HI 00 



bit 

in O 

vt-'o 

to cs 

co ci 3 

‘ to 

rt 

Hi 

r- ci 

in r^co 

-t 


o 

4-1 CO 


in O 

r^co 

cO O O 

0 

0 

Oco co 

t^O 

tn 

3 co 


• 

— 

W 

O' M 

CO 

to 

O' 44 

CO to f^H 

0 

*-4 

co 

in n 

0 ^ 

cn 

tn 



3 in 

to 

to to 

in O 

000 

0 


r-- 


co 

00 

co co 



4 -* 

rt 

3 - r-' ci 

O Cl 

COCO H 

0 

CO 

Hi 

3 Cl 

0 0 

HI 

CO w 


o 


co O' co m O 

to M 

m 3 

to 

rt 

CM 

co co 

'O CO 

O'CO 0 


o 

be 

O' o 

CJ 

co 3 - 

in 0 

0 



O 0 

in 3 

m 

Ol HI 


• 

c 

OO Hi 

CO 

10 


co to 

O M 

CO 

tn 

O' w 

CO 

tn 



C 4 

3 - in 

to 

to to 

to 0 

v. • 

000 . 

. O 




1^4 CO 

oo 

co co 




hi CT' hi 

to co 

Cl 0 

co in O 

O 

rt 

coco co 

co 3 

H 

M 3 .’ 



bic 

ci 

Cl 

3 - in 

3 M 

1^4 1-4 3 

to 

^t 

CM 

co co 

O' O 

O' 


o 

r^co 

0 

1-4 Cl 

co 3 

3 in m 

in 

to 

10 

rt rf 

CO Cl 

Cl 

O O' 


• 

G 

00 0 

co 

to 

O' M 

to m N 

■ O 

WH 

CO 

tn IN 

O' HI 

cn 

in O 



w 

3 “ m 

ID 

to to 

in 0 

OOO 

O 


1 ^ 


r -4 co 

CO 

co co 



4-1 

3 - 3 " t'' 

3 - co 

3 O' co >-4 co 

O m O 

Cl 3 

. .1 .. 

O co 

Hi 

0 0 



bjo 

O O 

0 

co 3 - 

co O O w co 

3 

-t 

Cl 

O' 3 

co O 

Hi 

G co 


© 

in O 

CO 

O' 0 

i-) CM 

ci co co 

CO 

co 

CO 

Cl Cl 

HI HI 

O 

O 


• 

G 

00 O 

(N 


O l-H 

C^ to IN 

0 

Hi 

co 

tn 


CO 

30 

.1 


w 

3 " m 

tn 

to to 

too OOO 

0 

n- 



t^co 

OO 

co co 



iJ 

r-'- O' ^ 

Cl Cl 

O Cl 

coco 0 

00 

OO O' 

H 

HI 

1^4 O' 


M 

bic 

CO 3 - 

0 

ci co 

Cl 0 O O co 

rt 

3 Cl 

O' 3 

CO H 

Cl 

M O' 


O 

Cl 3 - in 

co 

O' 0 

O H - 

HH 

H- 

Hi 

O O 

O' O'co 

tn 


# 

G 

co O 

Cl 

3-0 

00 Hi 

mo in 

Qn Hi 

CO 

in ( -4 

co O 

01 

30 



w 

3 " in 

to 

to to 

invO 

000 

O 

t'' 



co 

co 

co co 



4-1 

O' 3 - 

0 

0 Cl 

I'-' O 

co 3 3 

CO 


Hi 

O 3 

3 O' M 

CO Hi 


et 

bic 

O COCO 

Cl Cl 

1-4 O' in O co 

rt 

*t 

CO 

O m 

O' HI 

co 

Cl Hi 


© 

O Cl 

co 

in O 

f'- r^co O' O' 

0 

0 

O' O' co 

r^o 

in 3 


• 

G 

co 0 

Cl 

3 - <0 

co 0 

ci 3 O 

CO 

0 

CM 

30 

co 0 

Cl 

30 



w 

3 ' in 

10 

to to 

in O 

OOO 

0 




r^co 

00 

co co 



4 -> 

Cl co 

0 

CO Cl 

co co 

Cl 0 Cl 

co 

CO 

3 O 

Hi CO 

0 

O' CO 


H 

b£ 

ID Hh 

0 

O' M 

O co 

in 0 co 

rt 

rt 

CO O O 

O Cl 

~t 

CO Cl 


© 

co O 


Cl 3 

in in 0 r ^ 



r-s 

0 

O in 

3 CO Cl 


• 


r- 0 

Cl 

30 

co O 

Cl 30 

CO 

0 


3 O 

co O 

01 

30 



Cl! 

3 " m 

to 

to to 

in 0 

OOO 

0 




CO 

00 

co co 



4-1 

3 - ci 

Cl 

to w 

O' 1-4 

OOO 


00 

co tn 

CO 0 

0 

O O 


o 


co O 

in co O 

Oco 

3 O' Cl 

3 

3 

co O O 

O CO 

tn 

tn m 


© 

br 

vO CO 

O' 

O Cl 

Cl CO 

3 3 m 

to 

to 

to 

in 3 

3 CO 

Cl 

M C 


• 

c 

O >-i 

3 0 

co 0 

Cl -to 

CO 

0 

Cl 

30 

co 0 

Cl 

30 



W 


to 

to to 

in 0 

OOO 

0 




CO 

CO 

co co 


X 


0 »-i 

Cl 

CO 3 

in 0 

NOO O' 

0 

Hi 

Cl 

co 3 

in 0 

I'- CO O' 

hi>u 














C 

-G 


Cl Cl 

Cl 

Cl Cl 

Cl Cl 

Cl Cl Cl 

co 

co 

co 

co co 

co co 

co 

CO CO 

Cl 

.C 


HH M 

W 

hH H 4 

W H-t 

Hi HH HH 

M 

Hi 

Hi 

HI HI 

Hi Hi 

Hi 

t-H HI 





















































Table XVII. —reduction of barometric readings to feet. 


GUYOT'S BAROMETRIC TABLES. 


385 


o 

M 

01 

co f 

in O 

is CO 

O' 

O 

HH 

01 

co -t 

in o 

rsco 

O' 

O 

HH 

01 

en 




- 1 " 

-t 


f t- 


UD 

UT 

u~> 

ut in 

to IT) 

in m 

m 

o 

o 

vC 

o 

o 

HH 

HH 

HH 

HH r-i 

HH HH 

Hh HH 

HH 

HH 

HH 

HH 

HH HH 

HH HH 

HH HH 

HH 

HH 

HH 

HH 

HH 

HH 


g c . u 5 
o 7 

« Sw g 

OC 4 J .5 


«3 

-*.3 
•u o 
c c 

n •“ 

3 = 

o 

JZ ' 4H 

H° 


© 

C 


30 


J in -t M c i O O 

4 ;. 

v is o h cn m ts 

P- M K-, M HI 


r > t w co m ci o iO t<i 
m co mio 00 O h co in 


O m rs 
M CO t 


t m o rs 00 o 


Cl 


to cf mo Is 00 O' 


M Cl CO 


co O m rsi© cooo m ci o O m mco O m ci co ci m h-i q Oco is 


'*-00 co o co O' 
• m -t ci O co 
°® O' 1 — co in o 
® co O' O' O' O' 
£1 


OiMOHO O' M CO CO CO H 00 -t O' CO O 00 CO 00 IS 

vO t <N O rs 'tN OO CO o VO co 0"0 Cl CO t O o 

co C ci -t in is O' O ci i 1 O is O' O ci t m is 0 o 

O'OOOO O 'O M m M *h M m Cl Cl ci ci ci ci co 

MMMM MMIHMM yHMMMM 


hiO Cl ino t o tO O CO O t CO Cl TfiO IS IS Is O'O'O'O'O' 


'*-■ O' too O •-< 

■ COCI O O rs 

0 X 1 O' HH to TlO 

, c . co O' O' O' O' 

W 


>-> O coco 
in co O co in 
60 C oi co in 
O' 0000 


Ol tO O O 

co O I' t ^ 
is O' O ci t 
o O *—• i —1 *—• 


<t H Is Cl o 
co m hh co t 
in ts O O oi 

‘HI 'H H ’N ci 


O' H Cl Cl H 

O Is to O' m 

- 1 - in is co O 

01 ci oi ci co 


X* 

® 


01 01 O' co in in 01 Is m ci O fs coco CO O O hh co t iO hW 0*0 


h- hh vO O' 01 CO 
• ci o co rs m 

oc O' hh Cl -to 

*5 co O' O' O' O' 

W 


CO Cl O'O M 
CO H to O it 
co O m co in 
O' O O O O 


m is O' O' O 1 — t hh 0 O coinoOO 
HHCQinciOOcoOOco O m hh rs co 
tsco O oi co in rs O' O 01 co in ts co O 

O O 1 —< 1 —■ *—1 mmhhmm ci oi oi ci co 


M 

© 


© 


O co O ci m O t O m is o m ci co t co ci o co O 't'O to O ci 


" N t> h <t in 
. • O co rs m co 
OX) O' O o-l ^to 

,, co O' O' O' O' 


in t Cl co co 

H 0>tSTt 01 

co O' h co m 
O' O' O O O 


rs o 01 01 or 

O' Is t H OO 

O co O oi co 

O O M HI HH 


OootO"t 
in h co t 1 —i 
in ts co O oi 

HH HH HH Cl Cl 


ts O' O M O 

r> co o o ci 

co in rs co o 

01 M 01 01 CO 


-a 

CJ 


. o 

CO 

co O 


O m CO 

O CO 

cn 


HH 

co 

UT 

O m O 

COO 

HH 

t IS O 

co 

C 


S- f 

O' COO 

IS 

ts o t 

o o 

6 

co 

ir> 

UD 

m 

f M co 

co 


HH 

CO t 

in 



L-I 

•O 0 

o 

in co 

HH 

O' is in 

co O 

CO 

VO 

M 

O'O 

co O O 

C<“) 

o 

vC 

Cl co 

t o 


© 

b Xoo 

o 

Cl - 1-0 

IS O' M 

C^ UD 

o 

CO 

o 

HH 


m co 

o 

HH 

co 

m o 

OO 

O 



C co 

O' 

O' o 

o 

OOO 

o o 

o 

O 

HH 

Hh 

HH 

HH HH HH 

Cl 

Cl 

Cl 

Cl Cl 

Cl 

CO 

o 


W 




HH 

HH HH 

HH 

HH 

HH 

HH 

HH 

HH HH HH 

HH 

HH 

HH 

HH HH 

HH 

HH 

JC 


f co 

O Is 

c^ 

o rs'O 

cooo 

o o 

O oo 

vO 

M CO f 

o 

Cl 

CO 

Cl o 

O 


TJ 


4 -» 

'*-• U~) 

o 


o 

oco o 

cn co 

M o 

CO 

CO 

co 

is f m 

o 

HH 

4 

co 

Oco 

u 


. • o 

in 

cn hh 

o 

ud cn 

HH CO 

sO 

CO 

o 



hh co in 

HH 

co 

too 

Cl 

OO 

•c 

© 


o 

G -t Ul 

IS O M 

co -t 

o 

CO 

o 

HH 

CO 

iriO co 

O 

HH 

cn 

in O 

co 

o 

c 

• 

c co 

O' 

o o 

o 

OOO 

O O 

o 

o 

HH 

HH 

HH 

HH HH HH 

Cl 

Cl 

Cl 

Cl Cl 

Cl 

Cl 

3 


W 




M 

HH HH 

HH 

HH 

HH 

HH 

HH 

HH HH HH 

HH 

HH 

hh 

HH HH 

HH 

HH 

1 ! 





















CO 

4 -* * 

^ \Q 

bn<» 


t o m ci 


01 O O' HH 
CO h- O' CO 
O Ol co m 
O' O' O' O' 


ts O'co rs co 

m O 00 in m 
iO 't h O' Is 
rs (js h n t 
O' O' O O O 


O rsco to O t h 00 too m O m O m 


in co O 1-1 i- 1 

t HH O'O CO 
O CO O' H CO 
O O O hh hh 


Ooo t O t od h « con 

O O co O O 01 O' m « rs 

moco o h co to co O 

H H H Cl 01 01 01 01 Cl 01 


01 O' oi co h rsQi-iOoo oi imo is is o t ci O t ci 00 t O O 


CO 

tiS 


co co h co 
— O' co O 
O h com 
O' O' O' O' 


co co m co co 

t oi o rs m 

rs o' hi oi t 

O' O' OOO 


CO H CO t t CO H 00 cote 01 t'O fs cC 

01 O rs t h to in h co t hh rs co O' in 

O co O' m co t'O co O' m co t'O rs O' 

OOOi-ihh hh HH hh hh Cl oi oi oi oi oi 


in ^tco O O 


rs h co t n co hh in is co no m co 0 O'O to O n 


s-i O' m O' co m 
• O O' is o t 
OXcO Oh com 
C 00 CO O' O' O' 
W 


in m co O O 
01 O co iO co 
rs O' O 01 -t 
O' O' o o o 


O tO is rs o t h is 01 incoO^O 
HcomoiO OcoOOco O in 01 to t 

OisO'hoi t'O 00 O' h oi to I' o 

OOOh-iHh hh hh hh hh Cl 01 Cl 01 01 Cl 


co co 


Ttfsoo cO 01 in is is t O coo co O' O' O' co O O t ci O oo 


O O HH t'O 
• O' ISO -t 01 
Mis O' h co m 
C CO co O' O' O' 

W 


Is IS in 01 CO 
O co O t hh 
isco O ci t 
O' O' o o o 


coo O' O O 
O O co h co 
m is O' h oi 

O O O H. rH 


O' is —O m O' Oi t m t 
tncoinw is -t O O oi 

rf O Is O' h- 01 t'O Is O' 

H H H H 01 01 01 01 01 01 

H H H H H HH HH HH HH HH 


a G . tfl 

lu 

CO 


O hh oi co f imo noo O O h oi cot miOiscoO' Owoicof 
Ht-t-Ht't't in in in >n in in in in in m OiOOiOO 

J J . . ,_, ,_, 1 _to— l —4 —i —4 k _4 U_ <—I I—I H—I V—I —I 








































































Table XVII.—reduction of barometric readings to feet. 


3 56 


BAROMETRIC LEVELING 


a.s 

. CD 

tuo 


ioO 

InoO 

O' 

o 

k-H 

04 CO Tfr 

vn vo co 

O' 

O M 

04 OO Tf 

vn nO fN co O' 

~ l- 

c-g 


VO o 

© © © 




n- 

In I n 

fN 

CO CO 

00 00 CO 

CO CO CO OO CO 

03 Z> 

CQ V 

WJ 


HH 

k-H 

k-H HH 

k-H 

k-H 

k-H 

k-H HH 

M 

HH HH HH rH 

HH 

HH HH 

HH HH HH 

hh HH HH HH HH 

(D 
















sz 

c 

-C 

o 

c 

■4-* 

V 

cnco 

vO in 

rt- O 

O' m 

LO 

04 

k-H 

o- 



LO 

k-H 

O' Tf co COCO 

oi 't vn t>co 

04 In 

6 M 

HH 

co 

Tf fN H 

IT 04 Tf 

ct 

CD 

c 

fcu 



k-H 

k-H 

M 






HH HH 

HH 


3 

O 

JZ 

03 

Vh 


-t m>o Ingo 

O' 



HH 

04 co rf vnvO 

l^CO 

O' 

m oi cn 

H 

O 

















# 

in 

in 

IN O' H 

LO 

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O' 

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f 

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m© O oo in 


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04 

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k-H 

to 

vn o 

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be 

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o 

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o 

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k-H 

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co 

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co co 

CO 


H - H - H - rf 

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LO 

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vn m© 

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HH 

HH 

k-H HH 

k-H 

k-H 

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HH HH 

HH 

HH HH M HH 

HH 

HH HH 

HH HH HH 

HH HH HH HH HH 




O 

HH 

04 f fN 

04 

r>* 

CO M 

O 

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co 

r^co 

M O Tf 

Tf In hi O' O' 


CC 

v. 

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coco 

04 

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O M 

HH 

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fN 

04 lO 

O 04 Tf 

LO LO LO CO H 


• 

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co 

co 

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04 

hh 

O O Tf 

CO 04 © O Tf 


o 

be 

w 

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ir>vO 

CO 

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C4 rj- in 

fNCO O l-l 

to 

Tf VO 

IN O' o 

it <n Tf vO IN 


• 

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cn 

CO 

CO co 

CO 

CO Tf 

rr h- 

H- 

—f "T in in 

LO 

LO LO 

in m© 

vO O vO vO vO 



w 

M 

M 

k-H >H 

k-H 

k-H, 

k-H 

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u 

HH Hi HH k—1 

HH 

HH HH 

HH HH H-4 

HH HH HH Hi HH 



• 

04 

if)© O' CO 

O' vn 

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CO 

04 Tf OO Tf 04 

cn in m o t 



s-. 

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in 

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CO 

CO 04 

into O 

it it i-i o oo 



• 

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k-H 


C4 



01 fN 04 © 

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vn 0 

Tf oo co 

in m m O' 04 


o 

be 

M 

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k-H 

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Tf ©3 

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• 

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cn O'-) 

CO CO 

to 

CO *f 


O’ 

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LO 

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in m© 

vO vO 'nO sO 




HH 

k-H 

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l-H 

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HH HH HH 

HH HH HH HH HH 



. 

o 

in o *f 

O' 

o 

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fN O 

m oi h 

cn In 04 04 Tf 



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co 

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bio 


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in O' cn fN it 


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co rf© 

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04 

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co co co 

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LO 

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O vO vO vO 



u 

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HH 

k-H »-H 

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k-H 

M 

HH HH 

k-H 

HH HH HH HH 

HH 

HH k-H 

HH HH HH 

HH Hi HH HH HH 

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vO 

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04 

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HH 

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04 

CO© 

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on cn cn oi o 


bi: 

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04 fN 

it in O 

Tf CO 04 © O 

c 

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CO •Cf© 

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m u->© 

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k-H 

k-H 

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HH 

HH |-i 

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HH HH HH HH HH 



CO 

CO OO Tf 

k-H 

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vn >— co O O 

00 04 

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k-H 



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be 

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04 


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CO 04 In. 04 sO 

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04 © O Tf co 


c 

HH 

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04 CO 

LO 

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04 

Tf in 

fN 00 O' 

h 04 f in© 


• 

c 

oo 

CO 

co co 

CO 

CO 

H- 

O’ o* 

O’ 

Tf Tf Tf in 

LO 

LO LO 

LO LO LO 

O O O vO vO 

3 

*E 


w 

M 

k-H 

M k-H 

k-H 

k-H 

»-H 

HH HH 

HH 

HH HH HH HH 

HH 

Hh HH 

>-H HH HH 

HH HH HH HH HH 



# 

o o 

04 O' On 

O 

O O oo 

C4 

© too o 

O 

cnoo 

in Tf © 

O O f© o 


co 

V»-i 

HH 

co 

io o 

LO 

O' C4 

m© 

*f Tf oi O' m 

O Tf co tt cn 

LO LO LO Tf CO 


. • 

coco 

o- o 

LO 

O O 

i-i o 

Hh 

lO 1-4 O O 

LO 

O Tf oo cn fN 

IT in O' cn In 


c 

OJO 

HH 

04 

o-o 


O' 

O 

M CO 

LO 

vC CO O' m 

04 

Tf in © oo O' 

h oi cn in io 



w 


CO 

co co 

co 

co 


O’ o* 

O’ 

Tf -f Tf m 

LO 

LO LO 

LO LO LO 

vO O O vO vO 



M 

k-H 

k-H k-H 

k-H 

Hi 

k-H 

HH HH 

HH 

HH HH HH HH 

HH 

HH HH 

HH HH HH 

HH HH HH HH HH 




0* 

O'VO Tf 

CO 

C4 

CO 

Tt - co 

M 

fN Tf CO CO 

LO 

CO Tf 

04 04 Tf 

0"0 m r-' oi 


N 

Vi 

IT) 

04 

O' vo 

o 


O' o 

HH 

O O' fN Tf O 

vn o 

Tf fN O' 

O it m O O' 


bjb 

k—( 


oi co 


O' -f 

O' vn 

O 

in O' Tf O' Tf 

oo cn 

In H vn 

O Tf co oi vn 


o 

M 


O" vo 

in 

oo 

o 

H-. CO 

LO 

O r-r o O 

04 

co in o oo O' 

it oi cn w© 




to 

CO 

co co 

CO 

CO 

O" 

O’ O’ 

-+ 

Tf Tf Tf in 

LO 

LO LO 

LO LO LO 

O O vO lO O 




HH 

HI 

k-H «-* 

k-H 

k-H 

k-H 

HH HH 

HH 

k-H HH HH HH 

k-H 

HH HH 

HH HH HH 

HH hh HH hh HH 





04 

O co 

co 

CO 

0 

co 

04 

co vo mvo 

O' 

cn O 

O' O' 04 

oo © in co Tf 



V-i 

O' in 

O' O' O' 

CO 

C4 

Tf vn© 

in -f 04 O' vn 

w O 

O' 04 vn 

© IN fN © vn 



bio 

O' in 

k-H VO 



coco coco 

m oo c^ 

OI 

HH 

vn O Tf 

CO 04 o O tt 


© 

O 

04 

O' vo 


OO 

O 

HH CO 

~f 

vO rN O' C 

01 

co in© co O' 

O oi cn m© 


• 

G 

co 

CO 

co co 

co 

CO 

O" 

O* O’ 

O’ 

*f Tf -f m 

IO 

LO LO 

LO LO LO 

O vO vO O vO 



w 

HH 

HH 

k-H k—■ 

H—' 

k-H 

k-H 

k-H Hh 

HH 

H H H H 

HH 

Hh HH 

HH HH Hi 

—* *—1 1 >—1 HH 




O 

LO 

O- co 

"T 

LO CO 

HH O 

04 

O' co co C 

CO 

oo m 

vn vo O 

© vn in O' m 



4—1 

V-H 

CO 

k—< 

CO O' 

O' 

coo 

O' o 

HH 

O O' fN in 

HH 

\Q HH 

LO CIO HH 

oi cn cn oi it 


© 

• 

co 

t* 

O LO 

o 

O 

k-H 

vO b4 


C4 © h O 

hh 

LO O 

-foo cn 

In it in O' cn 


c 

bl 

O 

M 

CO LO 


OO 

o 

HH CO 

O’ 

O fN O' o 

01 

enm© NO 1 

O oi cn Tf so 


• 

G 

C*“> 

CO 

co co 

CO 

CO 

-t 

O’ O’ 


*f Tf Tf vn 

LO 

LO LO 

U> LO lO 

O O vO o o 



02 

HH 

k-H 

M k-H 

M 

k-H 

k-H 

HH HH 

HH 

HH HH Hh HH 

HH 

HH HH 

HH HH HH 

HH HH HH HH HH 

e.s 

. CD 

b/ojj 


vnO 

in oo 

O' 

o 

k—l 

C4 CO 

O’ 

in iO fN co 

O' 

O w 

oi cn Tf 

vn© mo O' 

\ *— 

c 

o o 

© © © 





NNNN 


oo oo 

oo oo oo 

oo co oo co co 

n: c 


k—( 

HH 

k-H k-H 

kH 

k-H 

HH 

HH HH 

HH 

HH HH HH HH 

HH 

HH HH 

HH HH HH 

HH HH HH HI k-H 

02 <u 



































































Table XVII. —reduction of barometric readings to feet. 


GUYOT'S BAROMETRIC TABLES. 


387 


M «r 


1/5 


•a 

c 

a! ■ 
x 

3 

O 

■C ' 

H 






















• C /5 

b£ <L> 


0 

HH 

Cl 

CO H - 

mo nco O' 

O M 

Cl CO t 

in 0 

co 

O' 

O M Cl CO t 

C XJ 


O' 

O' 

O' 

O' O' 

O' 

O' O' 

O' O' 

0 d 

0 0 

O 

O 

0 

0 

O 

O 

HH HH HH in 

Hi 

w 

C 


HH 

HH 

HH 

HH HH 

HH 

n HH 

HH HH 

Cl Cl 

Cl Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl Cl Cl Cl 

Cl 

-4 


1-1 

<U 

a/ 

t ao 

HH 

in O' 

Cl 


co 

O O' >- rf 

O 

coo 



Cl to OD 


0 

a 


in O 

co 

O' 6 

Cl 


HH 

Cl CO 

in 0 


O' 0 

HH 



H C! CO t 


a 


ti 




HH 

HH 







Hh 

HH 





a 3 

< 4 -H 

O 



t mo 

co 

O' 


HH 

ci co 

4 

5 

6 

1 ^ 00 

O' 



M Cl CO f 




. 

00 

0 

Cl 

coo 

Tf 

■f f ^ 

0 0 

co 0 

hh 

O' 

HH 


In O CO 

OOOO 

O' 


© 

<-»H 

Cl 

O' m 

0 *t 

CO 

M CO 

ir> O 

0 0 

UD 0 

HH 

O' in 

HH 


HH 

O O' Cl in vO 


• 

O' 

CM 

0 

O co 

0 

O COO O' 

ci m 00 m 

0 

v© 

O' C l 

0 

O' i-i •to 

CO 


© 

bje 

CO 

0 

HH 

CO O 

IT) 

co 

O' 0 

CM CO 

-f 0 

i^ 

O 0 

O' 

HH 

Cl 

c«-> 

■to 03 

O' 


* 

w 

0 

rs 




i'' 

co 

CO CO 

ao co 

co 

CO 

CO 

O' 

O' 

O' 

O' O' O' O' 

O' 



HH 

HH 

HH 

HH HH 

HH 

HH HH 

HH HH 

k-H HH 

Hi HH 

HH 

rH 

HH 

HH 

Hh 

HH 

Hi HH rH Hi 

HH 




Cl 

CO 

0 

00 Cl 

O 

HH IO 

ci n- 

co 0 

co co 

HH 

0 

HH 

HH 

U~) 

co 

O co cooo 

r-H 


X 

4-1 

S-h 

O' m 

HH 

O Hi 

in ao O 

ci co 

CO CO 

Cl rH 

O' 

0 


O' H- 

O' 

CO IS 0 M 

0 


• 


HH 

ud ao ci 

U) CO M 

UD CO 

HH 0 

t- O 

Cl 

10 CO 

O 

cn 

UD 

co O co in 



© 

bx 

00 

O 

HH 

ci -1- 

UD \Q CO 

O' 0 

Cl CO 

Tf-O 

r-. 

co 

O' 

HH 

CI 

CO 

tO is. ao 

O' 


• 

c 

0 






co 

ao co 

ao ao 

CO 

00 

CO 

O' 

O' 

O' 

O' O' O' O' 

O' 



w 

HH 

HH 

HH 

HH HH 

HH 

Hh hh 

HH HH 

HH HH 

HH HH 

HH 

HH 

HH 

Hi 

HH 

HH 

HH HH HH HH 

HH 




ID 

HH 

H- 

coco 

r^co co 

hh CO 

co 

0 IT) 

0 


UD 

IO 

0 

O' 

Cl O' 0 0 

O 


i- 

*-» 

v*-» 

UD 

M 

CO 

co r->. 

HH 

0 

O' 0 

d c 

O'co 0 

co O O 

CM 

0 

1-1 too O 

CM 


bi 

0 

0 

CO 

0 

t 0 


O to 

10 CO 

HH 

0 


O' Cl 

0 

fs O' l-l t O 


© 

CO 

0 

HH 

ci 0 

in O co 

O' 0 

Cl CO 

rf in 

L. 

co 

O' 0 

CM 

CO 

t m isod 

O' 


• 

c 

0 






CO 

co 00 

co ao 

ao 

CO 

co 

O' 

O' 

O' 

O' O' O' O' 

O' 



w 

HH 

HH 

HH 

HH HH 

HH 

HH Hh 

HH HH 

Hh HH 

HH 

Hi 

HH 

rH 

HH 

HH 

HH 

Hi ri Hi n 

l-l 




co 

IT) 

woo t 

CO 

in hh 

O Cl 

co 

0 t^O 

0 

CO 

0 ir> 

0 

00 0 00 t 

0 


© 

4 -> 

HH 

CO 

0 

O' 0 

CO 

m to r^- 

i'' 

to 

CO 

HH 


CO 

O' 

0 

co O' in co 

d 


u 

UD CO 

Cl 

in O 

CM 

O O' ci m 

CO HH 

0 

0 

cn 

U~) co 

0 

CO 

in co O ci 

m 


© 

OO 

O' 

HH 

ci co 

in 0 n> 

O' 0 

HH CO 

*t in 


CO 

O' 0 

Cl 

CO 

t in' is co 

O' 


• 

C 

0 

0 


r-~ 



C' co 

co co 

co co 

ao 

CO 

CO 

O' 

O' 

O' 

0 0 0 0 0 



w 

HH 

HH 

HH 

HH HH 

HH 

HH HH 

HH HH 

HH HH 

HH HH 

HH 

HH 

HH 

HH 

HH 

HH 

HI HH HI HH 

HH 




M 

O' 

O' CO O' 

O 

ci co 

CO HH 

co co 

HH CO 

O' 

C<“> 

CM 

0 

O 

0 

t Cl in h 

Cl 


1 ft 

t-i 

Vh 

CO 

0 

0 

O O 

0 

CO O 

ci -+ 

0 0 

Tf Cl 

O 

CO 

UD 

HH 

0 

Cl 

O 0 coo 

co ; 


tuc 

cn 


HH 

■f co 

HH 

0 CO 

*-* 0 

O 

COO 

O' 

HH 

0 

O' ci 

t O' i-i 

co 


© 

CO 

0 HH 

Cl CO 

in 0 

O' 0 

w CO 

t miO 

CO 

O' 

O 

Hi 

CO 

t in 0 co 

O' 


• 

c 

0 

0 





r^oo 

ao co 

CO CO 

CO 

co 

CO 

O' O' 

O' 

0 0 0 0 0 



w 

w 

HH 

HH 

HH HH 

HH 

Hi HH 

HH HH 

HH HH 

HH HH 

Hh 

HH 

rn 

HH 

HH 

HH 

HH HH HI HH 

HH | 




co 

Cl 

too m 

0 

O 0 

0 O 

r^-co 

Cl O 

Hi 

0 

UD CO 

0 

m 

O O' Cl O' 0 



+~i 

0 

HH 


<n 

HH 

in 

Hh 

HH HH 

i-i O co 

vn 

CM 

00 

0 

O' 

t fs w coo • 


** 

• 

Cl 0 

O COO 

O 

COO 

O' co 

vC O' ci in 


0 

e <"5 

UD CO 

0 

cn mao 0 

Cl 


© 

bl 

CO 

o> 0 

CM Cn 

in 0 ao O 

11 Cl 

t m 1O 

co 

O' 0 

«-H 

CO 

t m 0 ao 

O' 


• 

f— 

0 

0 



r^. 

t' 

co 

co ao 

co co 

co 

CO 

CO 

O' 

O' 

O' 

O' O' O' O' 

O' 



w 

HH 

HH 

Hi 

HH HH 

Hh 

HH HH 

HH HH 

HH Hi 

HH Hi 

HH 

HH 

HH 

HH 

HH 

HH 

HH HH HH n 

HH 


u 

c 

c 


</) 

.c 

w 

•a 

v 

u 

T3 

c 

3 


eo 

© 


N 

© 


© 


© 

© 


O 

o 


uicc n h 01 1^ t 10 O' M» m h n O O' Cl O' m in in O'O 


n n O''t 
. tOO M 'Cl 

Mao O' O N w 

c O O r-. rs 

W I— P-l HH M M 


00 h 'to co ao co Is in 
eo ci mco d t O coo 
•to r^co O ci t mo 
1^. is co co co co ao ao 


ci OO M is 

O'H TtNO 1 

r-s O' O m ci 
ao ao O' O' O' 


O' 

CO 


m in 00 
Cl t O O' 1- 1 
Tf mO O' 
O' O' O' O' O' 


O' O' N OO O COCOm COOO O OO "f CO m l-l Cl O CO in 1-1 11 O H" t-s 


v *-> o eo O m O 

• O' CO O -f 

MhO'O M co 
c 10 O is r>. 

W l-H 1-1 M M M 


t ao l -i co t in in in >t N 
O 'too coo O' M m 

t O Is- ao O h Cl rnimO 
r-' i~s r^co co ao co co co 


O N co 0"t 

co O co in ao 
O' O i- 1 Cl 
co co O' O O' 


O' COO O' >-> 
O co in r-> o 
it ino N O' 
O' O' O O' O' 


Cl Cl O CM Cl 


■t o O' h o o 00 m i- t m o co O r^co co m in 


s-i CO O O Cl I'' 

• co Cl in O' Cl 

bl O' O w co 

Coo rs t'' r-« 

W M M HH M l-H 


in c. o h ci n ci h O' 

O'M O O' ci m co m co 
in co O' i-Hdcomo 
co co co co co 


o 


I . —f M O Cl 

o O' ci t r-s 
co O m ci 
co 00 O' O' O' 


o O t o- 

O' ci to co 
co m o t^co 
0 0 0 0 0 


0 UD 


Or^r^. ot^OO'in mao mmO' Oco eon m co t O O' to 


0O 
oO 0 
oe r^* O' 

= 00 

W IH l-l 


coco co 

t h 

O I”* co 
rs 


ao m t O ao O' O' O' co O 
t CO *™i t I • O CO O O' Cl 
-t- m co O' w ci cc to 

cococococo 


t M CO t O' 
in co O co m 
r^co o 1- 4 ci 
co co O' O' O' 


too n t 
ao O comtN 
co mo r^co 
O' O' O' O' O' 


o h N cot 
O' O' O' O' O' 


1no fsco O' o H Cl 10 t mO i^oo O' 


O' O' O' O' o> 00000 

1 -^ M M M M d Cl N Cl Cl 


00000 

Cl Cl Cl Cl Cl 


O w Cl CO t 

M i-H M l-l t-l 

Cl Cl Cl Cl Cl 








































































Table XVII. —reduction of barometric readings to feet. 


388 


BAROMETRIC LEVEUNC 


flC.cn’ 
S'" bju 

P u r* XJ 


in 0 

hi M 

t^co 

hH hH 

O' 

hi 

O 

Cl 

H d CO 

ci ci d 

d 

m 0 

d d 

r^oo O' 
ci ci ci 

0 

CD 

hH d 

cd cd 

co 

Cl-' 

rt 

CD 

in 0 
ci co 

r^oo O' 

CD CD CD 



d 

d 

d 

Cl 

d 

Cl 

d d d 

d 

d 

d 

d d d 

d 

d d 

d 

d 

d 

d 

d d d 

CO 




















X 

• 4 - 

xi 


O 

d 

1 - r-. 

O 


hH 

CO 

1 - O 

co 0 

w 

d 





►H d 

C C 

05 • — 

£ c 

<U 

PL. 

O 

r^oo 

O' 0 

hH 


hH 

ci 

co 

*T 

mo co 

O O 

hH 





hi d 

C 

X 

05 


in 0 

i^co 

O' 


hH 

d 

3 

4 

in 0 !''• 

CO 

O' 





hH d 

h 

0 























a 

O' M 

r^-ao 

co 

Tt- O' CO 

CO 

HI 

in 

1- co 

001- 

1 - co 

CO 

CO 

1 - O' 0 


Ci 

C 

Vh 

CO 

CO 

O'co 

r^. 

O 

1 - H CO 

in 

hH O 

hH in O' 

COO CO 

O 

hH 

ci 

ci 

co d ci 


bi 

0 

CM 

1-0 

co 

O 

ci ~t m 


O' 0 

d co 1- 

0 

co 

O 

hi 

Cl 

CD 

1- in kO 


hH 

CM 

cd 

1 " L n 

00 O' O 

hH 

d 

1- in 0 r-H 

CO 

O' 0 

Cl 

CO 

1 - m 0 co 



w 

O 

0 

O 

0 

0 

O 

O O 

hH 

hH 


H H H 

hH 

hH Cl 

CM 

d 

d 

CM 

d d d 



CM 

d 

d 

Cl 

Cl 

Cl 

d d d 

d 

Cl 

d 

d d Cl 

d 

d Cl 

CM 

d 

Cl 

d 

Cl Cl Cl 




hH 

O' n 

00 

O' 

in O m w 

O 

in 

O 

O' 1- CO 


r-H d 

d 



CO 

1 - 0 d 


X 

v*-. 

0 

0 

0 

m 

’’T 

d 0 

CD 

O' m 

O' 1-00 

hH 


O' 0 

hH 

d 

ci ci hH 


bi 

c 

0 

hH 

cd 

LO 


O' 

w co 1-0 


O' 

O d co 

in 0 co 

0 

hH 

d 

co 1 - in 



0 

d 

cd 

~+ 

m 

0 

CO O' O 

hH 

CM 

co in 0 

CO 

O' 0 

hH 

CD 

1 - in 0 co 



w 

O 

O 

0 

O 

0 

0 

0 0 w 

hH 

hH 

hH 

f-H hH hH 

hi 

hi CM 

CM 

01 

Cl 

Cl 

Cl Cl d 



d 

d 

Cl 

CM 

Cl 

Cl 

d d d 

d 

d 

d 

Cl d Cl 

d 

d d 

d 

d 

d 

Cl 

d d d 



*-> 

Oco 

M 

CO 

0 

f^co -t 0 

0 

m 

m O O 


1- 0 

0 0 

0 

d 

1 - O co 


I'm 

Si 

to 

1 - in 

Tf 


Cl 

hH co in 

Cl 

co 

cd co cd 

0 

COO 

CO 

O' 

O 

hi 

hH hH O 



bi 

CO 

O 


-TO 

00 

O M CO 

in 

O 

CO 

O' hH d 

1- 

in O 

1 ^ CO 

O 

hi 

Cl CO rt 



0 

d 

CO 

-r 

m 

0 

CO O' O 

hH 

CM 

co 1 - O 

CO 

O' 0 

hH 

CM 

-f 

in 0 r-H co 



w 

0 

O 

O 

0 

0 

0 

O O •-> 

hH 

hi 

hi 

hi hH hH 

hi 

hH CM 

d 

d 

Cl 

Cl 

Cl d Cl 



CM 

d 

d 

CM 

Cl 

CM 

d d Cl 

d 

CM 

Cl 

d d d 

d 

d d 

CM 

Cl 

CM 

d 

01 CM 01 



- 4 -J 

co 


hH 

O' 

hH 

co 

O 7^00 



O' 0 0 0 

hH 

H NCO 

’T 

m 

d 

1 - hH 1 - 


<0 

'■•-4 

hH 

01 

CO 

Cl 

ci 

O 

O'O CO O 

0 

hH 

t"- hH in 

O' d 1-0 

CO 

O' 6 

odo' 


fci 


O' hH 

CD 

m 

t^CO O CM 

~T 

in 

co O hH 

d 

1 - in 0 


CO 

O 

hH d d 



0 

hi 

CD 

1- in 

0 

n- O' O 

hH 

CM 

CO 

N 

CO 

O' 0 

hi 

CM 

ID 

in 0 Ch co 



W 

0 

O 

0 

O 

0 

0 

O O 

hH 

hi 

hi 

hH hi hH 

hH 

hH CM 

Cl 

01 

d 

Cl 

d d d 

a: 


d 

Cl 

d 

Cl 

Cl 

Cl 

d d d 

d 

CM 

d 

Cl d d 

d 

d d 

Cl 

d 

d 

d 

Cl d d 

c 


j 



M 

O' 

Cl 

0 

d O M 


CO 

Tf in hH d 

r>.co in0 

CO 

CD 

hi 

1- hH 1 - 

c 

10 

Vh 

O' 

O 

hH 

0 

d 

O' 1 - d 

co 

1- 0 

in O 1- 


O co 

m 

ri 

06 

O' 

O' (Oco’ 

a 

bi 

U 1 CO 

O 

Cl 

1- 

in 

T'. O' 11 

CM 

1- 0 

O O' 0 

h- 

co 1- in 0 

1 ^ co 

O' O HH 

’ll 


0 

hi 

CO 

10 

vO 

r^co 0 

hH 

d 

co 

1- in 0 

CO 

O' 0 

M 

CM 

CD 

1- m noo 

0 


w 

0 

O 

O 

0 

0 

O 

O O W 

HI 

hH 

11 

hH hi M 

hH 

hi CM 

01 

Cl 

CM 

d 

CM CM CM 

CO 

X 


d 

CM 

CM 

CM 

CM 

CM 

d d d 

d 

Cl 

d 

d d d 

CM 

d d 

Cl 

d 

d 

d 

Cl d d 

+-> 

T3 



0 0 

hH 

0 

co 

hH 

td It 

hH 

d 

O' O r^co 

1-0 co 

m 

d 

’T 

hi 

1- d in 






















h. 



co 

O' O co 

r-> 

in co O 


coco 

1-co d 

0 

O' Cl 

1-0 

r^co 

co co 

'O 

c 


1 -vO 

CO 

0 

CM 

-tvO CO O 

hH 

co 1-0 O' 

O 

hi CD 


in 

vO 

to 0 0 


c 

O 

M 

d 


m 

cO 

f^CO 0 

hH 

CM 

CD 

1- in 0 

co 

O' 0 

hi 

d 

co 

“t 

m O CO 



w 

O 

O 

0 

0 

0 

0 

O O M 

hH 

hi 

hH 

hH hH hH 

hH 

hH d 

d 

d 

d 

d 

d d d 

£ 



d 

Cl 

Cl 

Cl 

CM 

Cl d Cl 

CM 

d 

Cl 

d d d 

Cl 

d d 

d 

CM 

d 

d 

d d d 




in 

m 0 

0 

1- 

0"0 


0 

CO 

in d 1- 

hH 

CO 0 

CO 0 

co 0 

1- d O 


co 


mo cO 

0 

in 

CD hH CO 

»n 

hH 


ci h 

in co hi 

CO 

in 

vO 


r^vd 


c 

• 

bi 

co 

m 


O' 

hH 

co 

m co 

0 

CM 

CD 

m rsco 

O' O d 

CD 


in vO 

co 0 


c 

O 

i—1 

CM 

CO 

in 

vO 

r-^co O' w 

CM 

CD 

1- in 0 


O' 0 

hH 

CM 

CD 

rt- m 0 



w 

O 

0 

0 

0 

0 

0 

OOO 

hH 

hi 

hH 

hH hi hH 

►H 

hH d 

d 

Ol 

Cl 

CM 

CM CM CM 



a 

d 

d 

Cl 

CM 

CM 

d d d 

CM 

d 

Cl 

d d d 

d 

d d 

d 

d 

d 

d 

d d d 




co 

1" O 

0 

m 

•+00 c- c 

co 

hH 

CO 

hH CO hH 


O co 

hH 

O' 

Cl 

0 

1 -COd 


c> 

si 

CD 


in 

in 

'-*■ 

CD 

1—1 O' r'- 

co 

6 

in 

M m O 

cd 

G O' ci 

CO 

in vO 

ID vO in 


0 

bi 

CM 

1-cO 

co 

0 

CM 

■t in 0 

O' 

hH 

CM 

1-iniN 

CO 

O' 0 

CM 

co 

m nco 


• 

a 

0 

hi 

01 

CD 

m 

0 

00 O' 0 

d 

CD 

1- mo 

r^oo O 

hH 

Cl 

CD 


in O t '- 



Cd 

0 

O 

0 

0 

0 

0 

000 

hH 

hH 

hH 

hH hH hH 

hi 

hH d 

CM 

d 

Cl 

01 

CM CM CM 



CM 

d 

CM 

CM 

Cl 

CM 

CM CM CM 

Cl 

Cl 

d 

d d d 

d 

CM CM 

d 

d 

d 

Cl 

d d d 




CM 

CO O 

0 

VO 

m 

0 O' CO 

hH 

in 

d 

0 O' 

1-0 in 00 


0 

O' 

1 - coco 




M 

d 

co 

co 

Cl 

hH 

cm in 

d 

co 


O' i- 00 

d 

m co 

0 

CM 

-t 

-t in in —r 


0 

bi 

hH 

co 

IO 


O' 

hi 

co 1-0 

CO 

O' 

hi 

d 1- in 

I"CO O' 

hi 

CM 

co 

“T 

m vC r-H 




O 

hi 

CM 

co 

-f 

0 

r-» co O' 

O 

hH 

co 

1- m O 

NX O' 

hi 

d 

CO 

Df- 

mo tN 



w 

O 

O 

0 

O 

0 

O 

OOO 

hH 

hH 

hH 

hH hi h-l 

hH 

hi hi 

Ol 

CM 

d 

01 

01 CM CM 



d 

d 

Cl 

Cl 

CM 

d 

d d d 

d 

CM 

d 

d d d 

CM 

d d 

d 

CM 

d 

01 

d d d 



4-) 

hH 

CO O 

HH 



d d O 

in 

O'O 

hH O' CO 

0 

co d 

0 

in 

O'CO 

CO COoO 


0 

V-t 

O' 0 

hi 

hH 

d 

O' DO vO CO O 

0 

d 

00 d G 

hi 


O' 

hH 

CM 

CD 

1" i- co 


© 


O' 

Cl 

—t O 

00 

O' 

-1 CD in 


CO 

O 

hH CO I" 

0 

1^ CO 

O' 

hi 

Cl 

CO 

D" in 



c 

O' 

hi 

Cl 

CD 

-f 

in 

r^co O' 0 

hH 

CD 

1- in 0 

J'-ao O' 

0 

Cl 

CD 

1- in O r-H 



w 

O' 0 

0 

0 

0 

0 

000 

hH 

M 

hH 

hH hH hH 

M 

hi hH 

Cl 

01 

d 

CM 

01 CM ci 



M 

01 

Cl 

Cl 

Cl 

d 

d d d 

d 

d 

d 

co d d 

d 

d d 

d 

d 

CM 

01 

d d d 

s.s 

• c /5 
bo a; 


in vO 

co 

O' 

0 

M d CO 


in 0 

1^00 O' 

O 

hH d 

CO 


in 

0 

Oh CO O' 

£ u 
rtiJ 
CC <u 

C X 3 


M 

hH 

hH 

hH 

h-» 

d 

d d d 

CM 

d 

d 

CM CM Cl 

co 

co ci 

co 

CO 

CD 

CD 

CD CD CD 

wj 


d 

d 

Cl 

Cl 

Cl 

d 

d d d 

d 

d 

d 

d d d 

d 

d d 

d 

d 

d 

d 

d d d 



















































Table XVII. —reduction of barometric readings to feet. 


GUYOT’S BAROMETRIC TABLES. 


389 


c S • 

g tD<U 
P V- C -5 
*t <u rT > o 
rt tf c 
03 « •= 


C/D 


'C U 
c G 

CT 3 

3 B 

O 

X 3 '+-< 

H° 


m d co tf ino in co O' O w d co tf >no inco O' O h n 


tf tf 
d d 


d 


tf 

d 


tf 

IN 


tf tf tf tf tf 

d d d d d 


m ui 10 in i/i 
d d d d d 


m min m in 
d d d <N d 


vO O O vO <o 

Cl d d d d 


J N to -tm m O In 
^ cn 4 m<d r> 06 O' 


O m M >h w d d d d 

m ci co tf uvo rvoo o 


O O O' 
• • • 
»-i d d 


tn rfmio 00 O' 


w <N co tf invo r^oo O' 


h n m 




00 

M 

co 

Cl 

0 

of in 

M 

CO 

M 


Cl 

CO 

O'O 

O' IN 

CO 

tf 

M 

ID 

tf 0 

M 

O' 

Ci 

SH 

6 

0"0 


w 


CO 

O' 


O' 

CO In O 

COO 

(X) 0 

ci 

CO tf 

Tf 

Tf 

Tf 

CO 

HH 

0 

bi) 


1 ^ CO 

O' 0 

0 


hH 

Cl 

Cl 

CO 

co 

•T* 

-f 

tf in 

ID 

ID 

Tf 

VO 

cO 

vo 

vo 

vo 


c 

O' 0 

f-H 

Cl 

’T 

UD O 

fN CO 

O' 

O 

M 

Cl 

CO *f 

IO sO 

IN CO 

O' 

O 

HH 

Cl 

CO 

Tf 


a 

Cl 


cn 

cn 

cn 

CO 

CO 

CO 

CO 

CO 

, T- 



'T ^ 

r T Tf 

Tf 

Tf 

tf 

ID 

VO 

VO 

VO 

vo 


Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl Cl 

Cl Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 


4 -i 

0 

cn 

M 


CO 

CO 

0"0 

co 0 

0 

O' 'tf 

CO CO 

NO 

M 

CO 

M 

ID 

tf 0 

Cl 

HH 

.08 


0 

co O 

co 

0 

id 

Cl 

co 

coco 

coo 

0 

coo 

00 d 

ci 

CO Tf 

Tf 

Tf 

Tf 

CO 

c 5 

h/'O O 

f^CO 

O' 

O' 

0 

O 

hi 

HH 

Cl 

Cl 

CO 

CO CO 

co tf 

Tf 

Tf 

Tf 

Tf 

Tf 

Tf 

-t- 

tf 

G 

O' 0 

M 

Cl 

CO 

TO 

co 

O 

0 

hH 

Cl 

CO -f 

UD 0 

In CO 

O' 

O 

HH 

Cl 

CO Tf 


H 

Cl 

cn 


co 

CO 

CO 

CO 

CO 

CO 

CO 

*1“ 

rt 

rt 


~t Tf 

tf 

Tf 

Tf 

IO 

vo 

10 

VO 

IO 


Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl Cl 

Cl Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 



W 

UD 

co 



Cl 

CO 

0 

CO 

Cl 

10 

10 

M 

CO M 

tf tf 0 

Cl 

0 

tf in 

M 

CO 

Cl 





























LO 

Cl 

O' 

0 

d 

CO 

toico 

Cl 

0 

0 

coo 

00 0 

Cl 

co 

Tf 

Tf 

Tf 

tf CO 

d 

0 

bio 


in 0 



GO 

O' 

O' 

O 

0 

l—t 

M 

Cl 

Cl CO 

Cl co 

CO 

CO 

CO 

co 

co 

CO 

CO 

CO 


O' 0 

►—1 

Cl 

co 

-f in cO 

CO 

O' 

0 

>—( 

Cl 

CO ^f 

in O 

r^co 

O' 

O 

M 

Cl 

CO 

Tf 


G 

tf 

Cl 


cn 

CO 

co 

CO 

co 

CO 

co 

CO 

r i“ 

’T' 


0- -r 

Tf Tf 

tf 

-f 

Tf 

ID 

VO 

vo 

vo 

LO 


Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl Cl 

Cl Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 


o 


d o in O O o bmtc n MMr^Ooo t) w w w o tf m m tf co 


. CO O tf d ©• 

be co tf m o o 
c o o 1-1 in co 
m m co co co co 

w N N N N N 


in m in d in 

in 00 co o o 

Tt - mo In co 
co co co co co 

d IN Cl W d 


d vO O co m 

O O O hi m 

OHMtOcJ- 

tf tf. tf tf Tf 

N N N M N 


00 O M CO Hi 
l-H d d d d 
m o in co o 

tf tf tf tf -f 

d d d d d 


ctiti-eon 

ci a m ci ci 

O M Cl CO tf 

in in in in in 

W N M M N 


X! 

o 


tr 

X 3 

_ _ 1 

T 3 

V 

u 

■a 

c 

3 

33 


>0 

c 


c 


■*-> 

Tf 

co 

00 

CO 

CO 

O' 

HH 

O' 

CO 

Cl 


TN 

tf 


VO 

O 

O 

IN 

0 

O' 

Tf 

vo 


tN 

vo 

CO 

HH 

CO 

Tf 

HH 

0 

Cl 

r-. 

HH 

vo 

O' 

d 

vo 

cd 

0 

HH 

cd 

cd 

Tf 

Tf 


Cl 

CO 

Tf 

VO 

vo 

vC 



CO 

00 

O' 

O' 

O' 

0 

0 

O 

HH 

*“H 

HH 

HH 

HH 

HH 


O' 

O 

HH 

d 

CO 

Tf 

vo 

0 


co 

O' 

0 

hi 

CO 

Tf 

vo 

0 


CO 

O' 

O 

HH 

r-r^ 

Cl 

CO 

CO 

CO 

CO 

CO 

CO 

co 

co 

co 

CO 

Tf 

Tf 

Tf 

~f 

Tf 

Tf 

Tf 

Tf 

Tf 

VO 

VO 

Uh 

d 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

d 

Cl 

Cl 

ci 

d 

d 

d 

d 

d 

d 

4 -) 

VO 

0 

O 

vo 

0 

CO 

VO 

CO 

|n 

In 

CO 

Tf 

O 

Tf 

CO 

CO 

O' 

VO 

O' 

co 

Tf 

VO 


H N lOi- 


vO m co O in 

be m ci co tf ~f 

- OO H N to 

r Tl d CO CO CO CO 

w N Cl N Cl Cl 


O O H lO 
in cO O in in 
' t mo in co 
CO CO CO no CO 
d d CO d d 


h m O Cl in 
00 co co C' O' 
O' O m Cl co 
co if if 't 
Cl Cl Cl Cl Cl 


In O' H d co 
O' O' o o o 
tf in in 00 O' 

tf ”f tf Tf tf 

Cl Cl N Cl Cl 


't t't cn ci 
OOOOO 
O H Cl CO tf 
m m in m m 
d d d d d 


CO 

© 


N 

© 


© 

© 




S-S 

O u X 
V. uo 

<2 W c 
CQ v — 


: co cl cl co O in O'co ci d co O in m O m in tf oc in to m cocn 


in tf ci O' in 
be O H Cl Cl CO 

c O' O m ci co 

(V! ci CO CO CO CO 
w Cl N N Cl Cl 


co O' m m o 
if if mo O 
•tmo in co 
co co co co co 
Cl Cl N Cl Cl 


O moo Cl in 

In I^CO OO 
O' O W Cl CO 

co tf tf tf tf 

Cl Cl IN Cl IN 


In O' "-i Cl CO 
CO CO O' O' O' 
^ miO Inco 

tf tf tf tf tf 

ci ci ci ci ci 


tf tf tf CO Cl 
O' O' O' O' O' 
O' O 1- 1 ci co 
tf in m in in 
N Cl Cl Cl Cl 


In co tf m co O co ci in /n cfo cooo n to m ci O in com coco 00 


. of CO M 0"0 

hr O' O '"I *™l Cl 

rCO O H Cl CO 

r T l Cl CO CO CO CO 

w Cl N Cl N Cl 


co O' <n o n 

co co of m in 

•t mo in 00 

co co co co co 

Cl Cl Cl Cl Cl 


O of CO H Tf■ 
•vO O O In tN 

O' O 1-1 Cl CO 

CO tf tf tf tf 

Cl Cl Cl Cl Cl 


IN O' M d CO 

in inco 00 00 
of in o in co 

of Of of of of 

Cl Cl Cl Cl Cl 


Of of Of CO Cl 
00 00 00 00 00 
O' O oh Cl co 
of in in in in 
ci ci ci <n ci 


O' m N toO co NO ci co O' Cl O "f m O n h mo ci m f co O' 


co ci O co m 

hVoo 0 0 0 1-1 

rCO O' M Cl CO 

M Cl Cl CO CO CO 

^ ci ci ci ci ci 


ci 00 of o m 

ci ci CO of '■f 

of m iO 00 
co co co co co 

N Cl N N Cl 


O' of 00 W Of 
f in miO iO 
O' O w Cl co 

COofofTfrf 

Cl Cl N Cl N 


N O' H N CO 
OO r-~ r-~ 
of m o co 
of Of -f -f of 
Cl Cl Cl Cl Cl 


of of Of CO Cl 
1^ 1' 

O' C 1-1 ci co 
of in in in in 
ci ci ci in ci 


Or^O'OO' N h h noo moo O h ci coMCi^fm ci m f O' O 


. CO H O' I'' of 
be 1^00 00 O' O 
boo o>0 h to 
H N ci co co co 

* Cl Cl Cl Cl Cl 


H 00 of O' Of 

H H d Cl CO 

of in >0 t '* co 

co co co co co 

Cl Cl Cl Cl Cl 


O' co r-' m of 
co -f of in in 
O' O H N CO 
CO of of of -f 
Cl Cl Cl Cl Cl 


co O' o ci to 
inunO'O'O 
of miO in 00 

of of of of of 
Cl Cl Cl Cl Cl 


of of of CO CO 

cO o o o o 
O' O | -i ci co 
of in in in in 
ci ci ci ci ci 


O h d co of m O in 00 O' OMCicoof invOlNooO' O h N tof 


of of of of Tf 
Cl Cl Cl Cl Cl 


of of of -f of 
Cl Cl Cl Cl Cl 


in in m m in 

Cl Cl Cl Cl Cl 


in in in in in 
Cl Cl ci ci ci 


O vO lO vO vO 

Cl Cl Cl Cl Cl 











































Table XVII. —reduction of barometric readings to feet. 


390 


BAROMETRIC LEVELING. 


n — bxiw 
o u G.C 
i- 41^1 y 
cfl *n W c 

03 « •= 


C /3 . 

J= jC 
~ u 
•a c 
c .5 
rt _ 
«j — 
3 aj 

H ° 


J3 

O 

c 

c 

cS 


C /3 

X! 

w 

-O 

4/ 

k- 

X 3 

C 

3 

B 


05 

c 


00 

© 


r- 

o 


© 

© 


10 

© 


inO Jsco O' O h 5 ) c<l rf inO isco O' O h n (O 't mO Isoo O' 


vO O O O vO 
N W N fl N 


xs is is is is 
d d d d d 


Is Is Is Is Is 

d d d d d 


CO 00 OO CO CO 

N N N N PI 


CO CO CO OO OO 
d d d d d 


.3 O' O' O' 03 00 CO 

QJ • • • • • • 

<u co rt m o is co 


O' O' co is is o m in -f 
O m pi cn 4 m O Goo 


O'co is 

O m ci 


it miO isco O' 


ci co -t m -o is co O' 


HH d CO 


• O - m ci in in n i-tOO'H O' m t^o O m h o 00 c ci in -t O' co 

1 , ,. - -* * ‘ . 

. O co O toO 
b£ in -t rf i-t -t 

e mo isco O' 

rj in in in in in 

H PI Cl Cl N Cl 


C CO 0 "t O 

CO CO Cl Cl Cl 

O 1-1 ci co -f 

cO O O O o 

Cl Cl Cl Cl Cl 


if O' O t> »i 

O O O' O' 
in o 1 s l '' co 

v -t *~3 cO lO 1^0 *03 

Cl Cl Cl Cl Cl 


rt N O' H O 
co is o O m 
O' O H d to 
O ts is i '' is 
d d d d d 


in o ts rs co 
-t co d hh o 
if mo tsco 

Xs ts is ts xs 
d d d d d 


in N itM co mao co coo in o co d ts O co ^to m M^j-coO'd 


Otoe too 
ti co to co co 
c mo rsco O' 
[j in in in m m 

” Cl Cl Cl Cl Cl 


o co O' m o 

d d M l- ^ 

O " d to-t 

cO sO 'O cO 'O 

Cl d d d d 


in o •h-co i-i 
O O 0300 co 
in o O is co 

00000 

Cl Cl Cl Cl Cl 


in |-n O Cl -t 
is O O m O" 
O' O M d CO 

O Is ts Is I-' 

d d d d d 


O is co co O' 
co d w o O' 

it mo ts is 
is is is is is 
d d d d d 


G rs o is o m 

'q O cd o TtM 

iso r, d ci d ci 

c mo tsco O' 

w m in m in in 

" Cl Cl Cl Cl Cl 


0 ci d 00 w ooococoood-tfco 0dco03d 


is rj- O in hh 

« t—1 ►—i O O 

O HI d CO i-f 

iC o 1O 0 o 

d d d d d 


O O rtco d 
O Otc h-fi 
*ct mo l''co 
iO iO iO iO O 
d d d d d 


in oh tom 
O in m rj- co 

O' O m d co 

O Is is ts Is 

d d d d d 


Is ao O' O' o 

d rH o O' O' 

it mo O l'' 
is ts is is x 

d d d d d 


J O H O' to "t 

o oo ot m 

fci) d M M HH M 

C imO NW O' 
ryi in in in m 10 

^ Cl d d d Cl 


dvOOcoiC in d m in O co co O d Cl 


co Ht O 
OOO 
O HI d 
O O iO iO o 
d d d d d 


lO M 
O' O' 
d CO 


iO m in O' co 
co co is o o 
it mo is co 
O iO ifl O iO 
d d d d d 


O O' m -to 
in t ^ to d 
O' O HH d CO 
O is rs is is 
d d d d ci 


co 

G 

1-1 

Ht 

X^ 

d 


— d CO d 

cto 6 h 
O O O'co 
in iO iO t '' 
X^ X^ x^. 
d d d d 


J O t*l H lO N OOOI^hh 

HH O' G Ht l-l 

bj h o o o o 

c mo r^co O' 

y inmininm 


d d d d d 


co in m 10 d 
O' O' O'co co 
O' O hh d co 
in iO iO O iO 
d d d d d 


hh r'- hh 1-1 

G HH o O CO 

r^iO o in 
it mic X'"co 
O O iO O O 

d d d d d 


o 

G 

-t 

O' 

iO 

d 


OtnOO NO H 00 d 


d in 
co d 

HH d 

x^ 

d d 


co 

x^ 

d 


CO 

O 

O' 

d 


O HH HH 

O O' CO 
in in o 
x^ 

d d d 


d 

x^- 

X'- 

t> 

d 


© 


co 

© 


d in -t CO O O' CO t M ^ o CO N |S CO IS rs -t IN CO in 00 N H 


. H-. ON^tCI 

bi o O' O' O' 

c in mo isoo 

o in m m in m 

^ ci ci d ci ci 


to m h n d 
to 00 00 ts Is 
O' O hh d co 
ino 'O'O'O 
d d d d d 


d O O 
vG sO in u~i *t 

nt imO Xs co 
vO O sO 'O iO 
d d d d d 


O omN 
co co d h o 
O' O hh d co 
0 NNNN 
d d d d d 


O' O d d co 
O' O'CO X's o 

co t mo is 

IS (S IN 

d d d d d 


j "t tsiO hh d dr^oosOO hh co d co O -t m d in o -tco O' n h 


hh O' is m ci 
bj) O'co co co co 
e nf m 10 x^ co 
m m in in in in 
^ d d d d d 


O' m hh is co 

r-s is x'n o O 
O' O HH d CO 
in sO iO sO 'O 
d d d d d 


co d is. hh in 
in in *t "t co 
rt mvO Js.00 
vO O O vO 'O 
d d d d d 


CO M HfO co 
d d HH O O' 

O' O hh d d 

O Is I~s X^ IN. 

d d d d d 


O hh d co Tt 

O'CO IsiO in 
co rt m O is. 

Xs. S Is Is Is 

d d d d d 


e* 

© 


j in co co ■ct'O 

**" w (jiGind 

hVcO Is X-N Is x~n 

c rt m o I's co 
jj in min in in 
w d d d d d 


O hh d O m sO COCO 0"0 O Cl O' CO -t dOCOOHn 


0"0 d co co 
iD'O'O mm 
O' O hh d co 
in o sC \0 O 
d d d d d 


co co fs hh in 
rt h^- CO CO Cl 
rj- in sO Xs. co 
vO iO G) O O 
d d d d d 


O' d rt X^ O' 
HH HH O O' OO 
O' O hh hh d 
o x^ i's is 
d d d d d 


h d co t m 
» NiO m t 

co -t in o i's 

Is Is Is is Is 
d d d d d 


© 

© 


Ac . tn 

C -HH hr II 
O k. 

rt hW B 
CQ 41 - 


j IS 0 HH 0 

O' 

O' mo 

0 

0 

h C"t in to 


OO 


d 

^ h 0 co in 

ci 

0"0 

Cl 

cn 


G coco ci 0 

O' d 

in co 

O 

i is is 0 O 

O 

XT) U~) 

m 

-t 

CO CO d d HH 

0 

0 

Oco 

CO 

2 ot in O xs co 

O' 0 

M 

d 

cn 

■t me is co 

O' 0 

O 

>—1 

<N 

] in id id in 

UD 

in 0 

0 

0 

0 

vO vO O O O 

0 



is 


d d d d 

d 

d d 

d 

d 

0 * 

d d d d d 


Cl 

d 

M 

d 

j co d co O 

d 

d co 

O 

O' 


O Ht O' Hh O' 

"i'O 

Ht 

0 0 

w 0 co in 

cn 

O vO 

coco 


O' Htco too 

O 

coo 

co 

M 

into O m in 

vr> 

in rt 


cn 

cn 

d d HH HH O 

0 

Oco 

IS 

IS 

- rt m O X'' co 

O' 0 

M 

d 

cn 

-x in 0 is co 

0 

0 0 

M 

d 

j ID in in in 

IT) 

in vO 

0 0 

0 

0 0 0 0 *o 

0 

0 




A d d d d 

d 

d Cl 

d 

d 

(N 

d d d d d 

d 

d 

d 

Cl 

d 

in 0 is oo 

O' 

O HH 

d 

CO 

rt 

in 0 tsco O' 

O 

W 

d 

cn 

r T 

O sO O sO 

0 



G 


G G G G G 

cd 

CO 

cd 

cd 

cd 

d d d d 

d 

d d 

d 

d 

Cl 

d d d d d 

d 

d 

d 

d 

d 


O m Xs in o 

d G nt in vC 
iso m t co 
co -t m o xs 

Is XS Is fs Is 

d d d d d 


O Ht *t in o 
• • • • • 
d ~t in o xs 
O m -x co d 
co Hf in O Is 

IS IS Is Xs Xs 
d d d d d 


m o Is qo O' 

co co co co co 

d d d d d 


























































Table XVII. —reduction of barometric readings to feet. 


GUYOT'S BAROMETRIC TABLES . 


39 



OQ v 


. C/3 


O m d co if 
• • • • . 

O' O' O' O' O' 
N N N N W 


in\D mo O' 

O' O' O' O' O' 
M d d d Cl 


O M d CO if 

6 o‘ o‘ 6 6 
co co co co co 


ino in co O' 

o 6 c o o 

CO CO CO CO CO 


01 

•S J 3 
■o 2 

c .S 

rt 

on c 

Feet. 

nO to if co d 

co -f to O In 

M 

OO 

8.6 

In O 

M ci 

If CO d 

CO If to 

O O' IN 

NO © IN 

2 >11 
•G O 


•f to O in co 

O' 

W 

d CO 

4 

5 

6 

In CO O' 

H 







• 


35 


M IN M d O' "t ICl CO O' n 


O' co co w oo O O co ci 


be 

c 


W 


co in r^o *f 
O' co in © to 
co O' O m d 
in in co co co 
d Cl d !M C) 


CO m O'© -f 
-t CO — O O' 
CO if to C © 
oo co co co co 
d d d d d 


i-i In if O © 

co © to if d 

r-^co O' O m 

co co co O' O' 

Cl Cl Cl Cl Cl 


M N N o H 

>-> O'CO © iO 
Cl Cl CO rf i/i 

O' O' O' O' O' 
Cl Cl Cl Cl Cl 


3 C 

© 


J H CO Cl CO H 

^ O' OO CO MD 
hfco in© to if 
r 00 O' O m Cl 
ryi in in co co co 
” N Cl ci d ci 


in in © ci ci 
*f ci O co in 

CO Cl HH 0-0 

co •t io ioo 

co co co co co 

Cl Cl Cl Cl Cl 


in ci in in © 

ci o in h in 

In l/l CO H 

In CO O O w 

co co co O' O' 

Cl Cl Cl Cl Cl 


ci i/i in coco 

co co co cd ci 

O CO IN in -f 
d d co if to 
O' O' O' O' O' 
Cl Cl Cl Cl Cl 


© 


Cl CO Cl If d 

v "“ o’ O' O' cd in 
b/OO © to if CO 
c co O' O m ci 
rv] (N NOO CO OO 
^ Cl Cl Cl Cl Cl 


in o\c/o if co 

in o h Cf © 
ci h o co in 
co “C io un © 
co co co co co 

Cl Cl Cl Cl Cl 


co m o c i O 

co O in cd O' 
O m co ci O 

fN CO O' O l"H 

co co co o O 

Cl Cl Cl Cl Cl 


in o h oo co 

A o A o tj- 
Oco nO *f co 
h m co 't in 
O O O O' O 
Cl Cl Cl Cl Cl 


© 

© 


O Cl O' CO m -t 

m o O d'cd 

he in no to co ci 

c oo o O w ci 

r-N in co co co 

^ Cl Cl Cl Cl Cl 


O M O IN I-H 

© mtoOco 
m O Oco nO 
co -f if >o no 
co co co go co 

Cl Cl Cl Cl Cl 


m O if© "f 
in h co it O 

IO if d H O 
In co O O i—i 
co co co O O 

Cl Cl Cl Cl Cl 


h imo 't O' 

NO I—I © M IO 

CO In in -f Cl 

h n co it in 

O O O O' O 

Cl Cl Cl Cl Cl 


JS 

o 

c 

c 

al 


o 

c/i 
J 2 
—< 
T3 
U 
u 
13 

c 

3 

X 




d 

O' if © 

m 

o 

cn 

d O' CO 

if d 

co 

O co 

© O' 

HH 

O' to 
















.05 


d 

HH M O 

O' 

CO 

© 

-f 1-1 O' 

© co 

O'© 

►H 

In d 

CO 

ci 

h/NO 

to -f CO 

l-H 

o 

O' CO In to 

If CO 

hH 

O 

O' 

vo 

- 1 * 

CO HH 


CO 

O' O M 

CI 

en 

cn 

if to © 

co 

O' o 

o 

HH Cl 

cn 

•f to 


w 


co co 

CO 

CO 

co 

co co co 

CO CO 

CO 

O' 

O' 

O' O' O' 

O' O' 


d 

d d d 

Cl 

M 

Cl 

d d Cl 

d d 

(N 

Cl 

Cl 

d d 

d 

d d 




O if IN © 

d 

in 

If d © 

in 

HH 


cn 

O if© 

If o 


















co 

nci H 

o 


to co O 

- 1 - 

hH 


CO 

O' f 

o 

-f O' 

bi 

m 

if co d 

►H 

O'Co 

o >n 

cn ci 

t —1 

O co 

© to 

cn 

d o 


co 

O' O HI 

01 

d 

cn 

if to © 

IN CO 

O' O' o 

Cl 

CO 

if to 


w 


in co co 

CO 

CO 

CO 

co co co 

co co 

CO 

CO 

O' 

O' a- 

O' 

O' O' 



d d d 

d 

d 

d 

d d d 

d d 

Cl 

Cl 

Cl 

Cl Cl 

Cl 

d d 


M 

© 


• Cl O io co in 


coin tN it O' O O' io co in 


it - O' h O © 


be 

c 


W 


if -t" co Cl M 
'ttOd M O 
CO O' O M Cl 
in in co co co 

Cl Cl Cl Cl Cl 


O CO NO if H 
O' in© to if 
ci co -tio'O 
co co co co co 

Cl Cl Cl Cl Cl 


O' to d co if 

C! H O CO IN 
In CO O' O' O 

co co co oo O' 

Cl Cl Cl Cl Cl 


O to i—i no O 

vO if CO M O 

m ci co -f in 

O' O' O' O' O' 

ci ci ci ci ci 


© 


ci O no oo O' toco 0"0 ci 


to to if co ci 
be co ci h O O' 
C co O' O HI M 
r T i (n in co co co 
^ Cl Cl Cl Cl Cl 


M O' 
CO NO 
ci co 


rN io co 
to if co 
to o 
co co co co co 

Cl d Cl Cl Cl 


co ci co ih m 

O in cd O no 
ci o Onco no 
fN oo co O' O 
co co co co O' 

Cl Cl Cl Cl Cl 


O' if to to h 


IO 


IN CI IN 

co ci o 
H ci co if 
O' O' O' O' 
Cl Cl Cl Cl 


Cl 

O' 

-f 

O' 

Cl 


U Cl M NO O' O 


© O w O' -f NO to Cl io IO 


co O' O O li 


nO nO to it it 
be ci m O O' co 
e'eo O O O - 
M in co co co 

Cl Cl Cl Cl Cl 


ci m o'O ~f 

In no "t CO Cl 
Cl CO if IO nO 
co co co co co 
ci ci ci ci ci 


HH CO IO M In 

HH O' CO IN to 
IN tNCO O' O 
CO CO CO CO O' 
Cl CI Cl Cl Cl 


coco if O' co 

f d hh o'CO 
H d CO to f 

O' O' O' O' O' 

Cl Cl Cl Cl Cl 


© 

© 


J d H NO H 
H*. . • • • • 

IN I— NO nO to 
hi IH O O' CO IN 

c co O' O' O i- 1 

M In in In 00 CO 

^ Cl Cl Cl Cl Cl 


OO d CO H In 

co ci O oo to 

© to if Cl HH 

ci CO -f to © 

co co co co oo 

Cl Cl Cl Cl Cl 


O' co to O' O' 

ci O' © ci co 
o co InnO if 

IN IN CO O' O 
CO CO CO CO O' 
Cl d Cl Cl d 


in co to to d 

-f o to o to 
co d O O' IN 
HH d CO CO If 
O' O' O' O' O' 
d d d d d 


o w d co if 

O' O' O' O' O' 
d d d d d 


IO 'O IN CO O' 

O' O' O O' O' 

d d d d d 


o M d CO If 

6 6 6 6 6 

CO CO CO CO CO 


to © IN CO O' 

6 6 6 6 6 

co co co co co 


















































392 


BAROMETRIC LEVELING 


Table XVIII. 

CORRECTION FOR r - r\ OR DIFFERENCE IN THE TEM¬ 
PERATURE OF THE BAROMETERS AT THE TWO STATIONS. 

This correction is negative when the attached thermometer at the upper 
station is lowest ; positive when the attached thermometer at the upper 
station is highest. 

(From Smithsonian Miscellaneous Contributions.) 


T — T' 

F. 

Correc¬ 

tion. 

T — T' 

F. 

Correc¬ 

tion. 

T — t' 

F. 

Correc¬ 

tion. 


T - T' 

F. 

Correc¬ 

tion. 


T - T ' 

F. 

Correc¬ 

tion. 

O 

Eng. ft. 

O 

Eng. ft. 

0 

Eng. ft. 


O 

Eng. ft. 


O 

Eng. ft.; 

I .O 

2-3 

21 .O 

49.2 

41.0 

96.0 


61.0 

142.9 


81.0 

189 7 

i -5 

3-5 

21.5 

50-4 

41.5 

97.2 


61.5 

144-1 


81.5 

190.9 

2.0 

4-7 

22.0 

51-5 

42.0 

98.4 


62.0 

145-2 


82.0 

192 . I 

2-5 

5-9 

22.5 

52.7 

42.5 

99.6 


62.5' 

146.4 


82.5 

193-3 

30 

7.0 

23.0 

53-9 

430 

IOO.7 


63.0 

147.6 


83.0 

194.4 

3-5 

8.2 

23-5 

55 • 1 

43-5 

101.9 


63-5 

148.8 


83-5 

195.6 

4.0 

9.4 

24.O 

56.2 

44.0 

103.1 


64.0 

149.9 


84.0 

196.8 

4-5 

10.5 

24-5 

57-4 

44-5 

IO4.2 


64-5 

151-1 


84-5 

197.9 

5 -o 

11.7 

25.0 

58.6 

45-0 

105.4 


65.0 

152-3 


85.0 

i 99 -i 

5-5 

12.9 

25-5 

59 7 

45-5 

106.6 


65-5 

153-4 


85-5 

200.3 

6.0 

14.1 

26.0 

60.9 

46.0 

107.8 


66.0 

154.6 


86.0 

201.5 

6.5 

152 

26.5 

62.1 

46.5 

108.9 


66.5 

155-8 


86.5 

202.6 

7.0 

16.4 

27.0 

63.2 

47.0 

IIO. I 


67.0 

i 57 -o 


87.0 

203 8 

7-5 

17.6 

27-5 

64.4 

47-5 

hi.3 


67.5 

158.1 


87.5 

205.0 

8.0 

18.7 

28.0 

65.6 

48.0 

II2.4 


68.0 

159-3 


88.0 

206.1 

8-5 

19.9 

28.5 

66.8 

48.5 

113.6 


68.5 

160.5 


88.5 

207.3 

9.0 

21 . I 

29.0 

67.9 

49.0 

114.8 


69.0 

161.6 


89.0 

208.5 

9-5 

22.3 

29-5 

69.1 

49-5 

116.0 


69 5 

162.8 


89-5 

209.7 

IO.O 

23-4 

30.0 

70-3 

50.0 

117 • I 


70.0 

164.0 


90.0 

210.8 

TO.5 

24.6 

30-5 

71.4 

5°-5 

118.3 


70.5 

165.2 


90.5 

212 O 

II .O 

25.8 

31-0 

72.6 

51.0 

H9-5 


71 .O 

166.3 


91.0 

213-2 

11 -5 

26.9 

3 1 • 5 

73-8 

5 1 -5 

120.6 


7 i -5 

1675 


9*-5 

214.3 

12.0 

28.1 

32.0 

75 -o 

52.0 

121.8 


72.0 

168.7 


92.0 

2, 5-5 

12-5 

29-3 

32.5 

76.1 

52.5 

123 0 


72-5 

169.8 


92. s 

216.7 

13.0 

30.5 

33 -° 

77-3 

53 -o 

124.2 


73 -o 

171 .O 


93 -o 

217.9 

13-5 

31.6 

33-5 

78.5 

53-5 

125.3 


73-5 

172.2 


93-5 

219.0 

14.0 

32.8 

34 -o 

79.6 

54 -o 

126.5 


74.0 

173-4 


94.0 

220.2 

M -5 

34 -o 

34-5 

80.8 

54-5 

127.7 


74-5 

174-5 


94-5 

221.4 

15.0 

35 -i 

35 -o 

82.0 

55 -o 

128.8 


75-o 

175-7 


95.0 

222.5 

15-5 

3 6 -3 

35-5 

83.2 

55-5 

130.0 


75-5 

176.9 


95-5 

223.7 

16.0 

37-5 

36.0 

84-3 

56.0 

131.2 


76.0 

178.0 


96.0 

224.9 

16.5 

38.7 

36.5 

85.5 

56.5 

132.4 


76.5 

179.2 


96.5 

226.1 

I7.O 

39-8 

37 -° 

86 7 

57 -o 

> 33-5 


77.0 

180.4 


97.0 

227.2 

17-5 

41.0 

37-5 

87.8 

57-5 

134 7 


77-5 

181.6 


97-5 

228.4 

18.0 

42.2 

38.0 

89.0 

58.0 

135-9 


78.0 

182.7 


98.0 

229.6 

18.5 

43 3 

38.5 

90.2 

58.5 

i 37 -o 


78.5 

183.9 


98.5 

230.7 

19.0 

44-5 

39 -o 

91.4 

59 -o 

138.2 


79.0 

185.1 


99.0 

231.9 

* 9-5 

45-7 

39-5 

92.5 

59-5 

* 39-4 


79-5 

186.2 


99-5 

233-1 

20.0 

46.9 

40.0 

93-6 

60.0 

140.6 


80.0 

187.4 


100.0 

234-3 

20.5 

48.0 

40-5 

94.9 

60.5 

141.7 


80.5 

188.6 


100.5 

235-4 




































Table XIX. 

CORRECTION FOR THE DIFFERENCE OF GRAVITY AT VARIOUS LATITUDES. 

Correction positive from latitude o° to 45 0 ; negative from 45 0 to go°. 

(From Smithsonian Miscellaneous Contributions.) 


GUYOT'S BAROMETRIC TABLES 


393 


1 

1 ) 

u n 

£ Q 

0000 

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vcT 06 

8 8 8 8 8 
m cn Mr tr 

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w 


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394 


BAROMETRIC LEVELING 


Table XX. 

CORRECTION FOR DECREASE OF GRAVITY ON A VERTICAL. 

(From Smithsonian Miscellaneous Contributions.) 


Approxi¬ 

mate 

difference 
of level. 

Decrease of grav¬ 
ity on a vertical. 
Positive. 

0 

4- 500 

Eng. feet. 

Feet. 

Feet. 

1,000 

2.5 

3-9 

2,000 

5.2 

6.6 

3,000 

7 • 9 

9-3 

4,000 

io.8 

12.2 

5,000 

*3 7 

15.2 

6,000 

16.7 

18.3 

7,000 

19.9 

21-5 

8,000 

23.1 

24.7 

9,000 

26.4 

28.1 


Decrease of grav- 


Approxi¬ 

mate 

difference 
of level. 

ity on a vertical. 
Positive. 

0 

+ 500 

Eng. feet. 

Feet. 

Feet. 

10,000 

29.8 

3 i -5 

11,000 

33-3 

35 -i 

12,000 

36-9 

3 8 -7 

13,000 

40.6 

42.5 

14,000 

44-4 

46-3 

15,000 

48.3 

50-3 

16,000 

52.3 

54-3 

17,000 

56.4 

58-4 

18,000 

60.5 

62.6 


Approxi¬ 

mate 

difference 
of level, 

Decrease of grav 
ity on a vertical. 
Positive. 

0 

+ 500 

Eng. feet. 

Feet. 

Feet. 

19,030 

64.8 

67.0 

20,000 

69.2 

71.4 

21,000 

73-6 

75-9 

22,000 

78.2 

80.5 

23,000 

82.9 

85.2 

24,000 

87.6 

90.0 

25,000 

92.5 

94-9 


Table XXI. 

CORRECTION FOR THE HEIGHT OF THE LOWER STATION.— 

POSITIVE. 


(From Smithsonian Miscellaneous Contributions.) 


Approximate 
difference 
of level. 

Height of the barometer, in English inches, at lower station. 

16 

18 

20 

22 

24 

26 

28 

Eng feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

1,000 

1.6 

i -3 

I .O 

0.8 

0.6 

0.4 

0.2 

2,000 

3 • 1 

2.5 

2.0 

i -5 

1.1 

0.7 

o -3 

3,000 

4-7 

3-8 

3 -o 

2.3 

i -7 

1 . 1 

o-S 

4,000 

6-3 

5 -i 

4.0 

3 -i 

2.2 

*•4 

0.7 

5,000 

7.8 

6.4 

5 -o 

3-8 

2.8 

1.8 

0.8 

6,000 

9-4 

7.6 

6.0 

4.6 

3-3 

2 . I 

I .O 

7,000 

II .O 

8.9 

7 -i 

5-4 

3-9 

2.5 

1.2 

8.000 

12.5 

10.2 

8.1 

6.2 

4-4 

2.8 

1 - 3 

9,000 

14.1 

n.4 

9.1 

6.9 

5 -o 

3-2 

1 5 

10,000 

I 5*7 

I2.7 

IO. I 

7-7 

S • 5 

3-5 

i *7 

11,000 

17.2 

14.0 

II . I 

8-5 

6.1 

3-9 

1.8 

12,000 

18.8 

1 5-3 

12.1 

9.2 

6.6 

4-2 

2.0 

13,000 

20.4 

16 -5 

* 3 -i 

IO.O 

7.2 

4.6 

2.2 

14,000 

21.9 

17.8 

14.1 

10.8 

7-7 

4.9 

2-3 

15,000 

23-5 

19.1 

I 5 -I 

11 -5 

8-3 

5-3 

2.5 

16,000 

25.1 

20.3 

16.1 

! 2-3 

8.8 

5-6 

2.7 

17,000 

26.6 

21.6 

17.1 

i 3 -i 

9.4 

6.0 

2.8 

18,000 

28.2 

22.9 

18.1 

13.8 

9.9 

6-3 

3 -° 

19,000 

29.8 

24.1 

19.2 

14.6 

10.5 

6.7 

3-2 

20,000 

3 1 -3 

25.4 

20.2 

15.4 

II .O 

7.0 

3-3 

21,000 

32.9 

26.7 

21.2 

16.1 

11.6 

7-4 

3-5 

22,000 

34-5 

28.0 

22.2 

16.9 

12. I 

7-7 

3*7 

23,000 

36.0 

29.2 

23.2 

17.7 

12.7 

8.1 

3-8 

24,000 

37-6 

30-5 

24.2 

18.5 

13.2 

8.4 

4.0 

25,000 

39 -* 

31.8 

25.2 

T9.2 

13.8 

8.8 

4- 1 















































ANEROID BAROMETER. 


395 


174. Aneroid Barometer. —This instrument depends for 
its operation on a shallow cylindrical metal box, the top of 
which is made of corrugated metal and is so elastic as to read¬ 
ily yield to changes in the pressure of the atmosphere. The 
interior of this box is exhausted of air , so that when the 
atmospheric pressure increases the top is pressed inwards, and 
when it decreases the elasticity of the corrugated top moves 
it outwards. These movements are transmitted by multiply¬ 
ing levers, chains, and springs to an index which moves over 
a scale. (Fig. 114.) 

Aneroids as made by various instrument-makers differ in 
the mechanism employed to multiply the linear motion of the 
end of the vacuum index and in the arrangement of figures 
on the face of the scale. The instrument is graduated by 
comparing its indicator under different pressures with those 
of a mercurial barometer, and is tested in a vacuum pump, 
and a scale of correction is usually prepared with a view to 
making it independent of temperature changes. At the back 
of the instrument is a screw which presses against the end of 
the vacuum box so that it may be adjusted at any base 
elevation. 

The scales on the face are usually two in number, one for 
inches of atmospheric pressure, and the other for altitude in 
feet. The scale of feet is frequently made movable so that it 
may be set at a known altitude opposite to the index pointer, 
after which changes in the index hand will indicate relative 
changes in altitudes as based upon the setting. 

175. Errors of Aneroid —The aneroid is very convenient 
as a movable instrument, requiring no time to place it in po¬ 
sition for observing, as does the mercurial barometer, and being 
at all times in condition for immediate and direct reading, as 
is a watch. It is inferior, however, as a hypsometric instru¬ 
ment to the mercurial barometer, chiefly because it is subject 
to the following sources of error: 


396 


BAROMETRIC LEVELING. 


1. The elasticity of the corrugated top of the vacuum 
chamber is affected by rapid changes in pressure. 

2. Its readings are affected by changes in temperature 
which it is impossible to readily compensate. 

3. The different spaces on the scale are seldom correct 
relatively one to the other, but the scale of pressure or inches 
is more accurate than the scale of feet, since the latter con¬ 
tains the errors due to the formula by which it was graduated. 

4. The weight of the instrument affects its indications, its 
readings differing in accordance with the position in which it 
is held. 

5. It lacks in sensitiveness, frequently not responding 
quickly to changes of altitude. 

6. The chain and levers sometimes fail to quickly respond 
to the movements required of them. 

7. Because of its containing so many mechanical parts 
these are subject to shifting or jarring by movement made in 
transporting it, the only remedy for which is frequent com¬ 
parison with known altitudes or a mercurial barometer. 

The aneroid is not an instrument of precision, and the least 
reading which it is capable of is about 0.025 of an inch, corre¬ 
sponding to nearly 25 feet, and no system of verniers nor 
multiplying scales will increase the precision. The range of 
pressure of the aneroid is limited, and if used for a greater 
altitude, or a pressure lower than that within its range, the 
spring runs down ; in other words, the spring ceases to act 
after the pressure has been lowered too far. 

It frequently happens, as on the approach of a storm, or 
change from stormy to clear weather, that atmospheric pres¬ 
sures will change in a few hours by over an inch. This means 
an apparent change of elevation at the same place of 1000 
feet or more. (Art. 168.) 

176. Using the Aneroid. —The aneroid barometer is used 
very extensively in the topographic surveys executed by the 
U. S. Geological Survey, Excepting in country of very flat 


USING THE ANEROID . 


397 


slopes, it has been used almost exclusively by that organi¬ 
zation in sketching contours over an area of 800,000 square 
miles which have been already mapped. As previously 
stated, the aneroid has been found to be erratic and unreliable 
for exact work. It has also been found that where properly 
handled and attention is paid to its eccentricities it is a suffi¬ 
ciently accurate instrument to permit of sketching contours 
of intervals not less than 20 feet, in moderately rolling country, 
with all the accuracy necessary for a scale of one mile to the 
inch, and in very mountainous country for even larger scales. 

The topographers of the U. S. Geological Survey carry 
• aneroids of the simplest form , unencumbered by verniers and 
similar in size and general appearance to that shown in Fig. 
114. These instruments are from 2 to 2 \ inches in diameter, 
and aneroids of various ranges are employed according to the 
altitudes of the country under survey. In a region in which 
the heights do not exceed 2000 feet a 3000-foot aneroid is 
carried. When the altitudes exceed 3000 or 4000 feet a 
5000-foot aneroid is carried. An aneroid cannot be used 
with any safety in determining heights which approach nearly 
to its range. 

The instrument is carried loosely as a watch in the pocket. 
The slight jolting which it thus receives in riding or walking 
is just sufficient to keep the needle from sticking and aid it 
in responding to the changes of altitude. In reading it it 
should invariably be held in the same position. Some prefer 
to hold it horizontally, the better way, however, is to hold it 
vertically in front of the eye, suspended by the carrying-ring. 
In reading it the eye should always be held in the same 
position with relation to the needle, to avoid the effect of 
parallax , and the case of the aneroid should be rapped gently 
but sharply in order to loosen the spring or needle should 
either stick, such rapping being more effective if performed 
with a hard substance, as the finger-nail or lead-pencil, than 
with the fleshy part of the finger. 


393 


BAROMETRIC LEVELING . 


In setting out to work, the reading of the aneroid should 
be noted to see if it has changed materially from the reading 
noted in camp on the previous day. This gives some indica¬ 
tions of the condition of the atmosphere. Before starting out 



w 

< 


6 

i-t 

fu¬ 


tile sliding foot-scale of the aneroid should be revolved so that 
the hand shall point to the altitude of the camp or other 
known elevation near the starting-point. On arrival at the field 
of work the aneroid should be again read at some point the 
elevation of which is known, and the effect of atmosphere 






































































































































USING THE ANEROID. 


399 


or sudden change in height in preventing its recording this 
known elevation correctly, as compared with that at which it 
had been previously set in camp, should be noted. If the 
aneroid appears to have acted erratically, it should be used with 
great care at first and frequently checked, lest the atmospheric 
conditions be unsuited to its use. There often occur days 
on which it is impossible to use the aneroid (Art. 175), 
when results are desired which permit of sketching contours 
of intervals as small as 20 feet. On such days the topogra¬ 
pher should either obtain numerous check elevations by ver¬ 
tical angulation, or should confine his route to sketching where 
elevations already obtained are numerous, or should devote 
himself to plane-table triangulation, office work, or some other 
phase of his duties. 

If the aneroid seems in proper condition for use,—and this 
is best verified by carrying two aneroids, lest one for some 
reason be out of order,—the topographer may proceed to 
sketch contours by it (Arts. 13 and 17). In this work he 
should use only one of the aneroids, that which seems in best 
condition, making no attempt to check it by the other, or to 
take a mean of the readings of the two, but depending for 
such adjustment on checking it by elevations obtained by 
better methods. Setting the aneroid at his starting-point 
and at a known elevation, he drives over the roads, consulting 
it to determine the height at each contour crossing of the 
route traveled. He may rely upon it for sketching contours 
of small interval for distances not exceeding a couple of miles 
without rechecking it. Where the changes of slope are not 
great and the journey is made with considerable speed, as by 
driving in a vehicle, and where the time consumed in travel 
is comparatively short, the aneroid may be safely used for 
distances as great as three to five miles, though in such cases 
it may not check out within a contour interval on the next 
comparison, when a portion of the journey just made must be 
retraveled and the topography resketched. Where the con- 


400 


BA ROME TRIC LE VELING. 


tour interval is greater than 30 feet, as 50 or even 100 
feet, longer journeys may be made and greater differences of 
altitude encountered without introducing errors in the aneroid 
reading which will equal a contour interval in amount. 

In using the aneroid in the above manner its exact reading 
should frequently be marked on the map, especially at all 
junction points of roads and trails. Accordingly, as the 
topographer in driving or walking over the various roads or 
trails comes back to one of those points at which he has 
already noted the aneroid height, thus closing a circuit , he 
adjusts his aneroid by comparison with the recorded height 
as though he were adjusting it on an elevation obtained by 
better methods. In this manner he may be able to extend 
the range of use of the aneroid by throwing out closed circuits 
of aneroid elevations one from the other to distances as great 
as four or five miles to the next elevation of first quality with¬ 
out introducing errors beyond his contour interval. Such re¬ 
sults can only be obtained under the most favorable atmos¬ 
pheric conditions. 

In using the aneroid in geographic or exploratory surveys 
where frequent checks cannot be had on known elevations, or 
by closing back on aneroid heights already recorded, the in¬ 
strument must be handled in a different manner. It must 
still be used with the same care, and the beginning of the 
journey must be made in the same manner. Immediately 
upon making a stop for a rest or overnight, or for an interval 
of time of even five minutes, the height indicated by the 
aneroid should be at once recorded. In starting out again the 
aneroid will be again read, and if the elevation which it 
records has changed, the scale must be reset to that noted 
when the stop was made. To get the best results the journey 
should be made as rapidly as possible from one stopping-point 
to the next. 

As already stated (Art. 175), the aneroid acts sluggishly 
upon making any sudden change of elevation which is consid- 


USING THE ANEROID. 


401 


erable in amount. Thus in ascending or descending a high 
and steep hill the aneroid will fail to record the full altitude 
passed over if read immediately upon arrival. It should be 
so read, however, but a stop of a few minutes should be 
made at the top or bottom of an inclination, and thereafter 
the aneroid be again read, in which case if not affected by 
unaccountable atmospheric conditions it will have responded 
gradually to the change of elevation and will note an increased 
difference of height. Frequent comparison with known eleva¬ 
tions in the conduct of aneroid work has shown that the amount 
of change by which the record of the aneroid is too small varies 
from 2 to 5 per cent, according to the speed with which the 
journey has been made, the condition of the aneroid itself, 
and the difference of elevation. It is therefore safe to add 
this amount to or subtract it from the record of the aneroid, 
as noted upon immediate arrival at the top or bottom of a 
high, steep slope. The scale of the aneroid, however, should 
not be corrected for this difference, since the aneroid will 
gradually come back itself to the change of elevation which 
it should have originally noted. 

In railway and other topographic surveys in Germany 
even more faith is placed in the results of aneroid readings 
than the most firm believers in the instrument in this country 
would advocate. Mr. F. A. Gelbcke states that a careful ob¬ 
server is able to reach an approximation of from three to six 
feet of elevation with certainty. Such a high degree of accu¬ 
racy is obtained of course only where the aneroid is frequently 
checked by reference to spirit-level elevations, as in making 
a topographic survey for railroads, where a base line is lev¬ 
eled through and the aneroid is used at comparatively short 
distances and for small changes of elevation. 

Calculations of heights from such observations are made 
graphically, the aneroid readings, after correction for tem¬ 
perature, being plotted on cross-section paper. On this, with 
the aid of the barographic notations and the readings at the 


402 


BAROMETRIC LEVELING. 


bench-marks and other check stations, a horizontal curve is 
constructed. This is an a 7 ieroid diagram , from which it is 
only necessary to read the ordinates of the curve at the 
stations, with a scale varying to suit the observed changes of 
temperature, in order to obtain the elevations of the stations. 
Thus the desired heights are furnished without calculation 
and in the least time, and so that large errors in determina¬ 
tion of the elevations are practically excluded. 

177. Thermometric Leveling. —Differences in elevation 
may be ascertained with a certain degree of approximation 
by means of determining the boiling-point of water. This is 
because when water is heated the elastic force of the vapor 
produced as it is transformed into steam increases until it 
becomes equal to the incumbent weight of the atmosphere ; 
this pressure then being overcome, the vapor bursts into 
steam. It is evident, therefore, that the temperature at 
which water boils in open air depends upon the weight of the 
column of atmosphere above it, and this fact is made use of 
in determining the differences of altitude. 

The temperature at which water boils under different 
pressures has been determined by experiment. It is only 
necessary, therefore, to observe the temperature at which 
water boils at any place, and by referring to a table to find 
the corresponding height of the barometer or elevation 
above the sea. Account may be taken of the effect of varia¬ 
tions in temperature, moisture, pressure, etc., but the errors 
inherent in the method itself are so great as to make such 
attempt at refinement of little value. Table XXII gives the 
approximate elevations above mean sea-level for different 
temperatures Fahrenheit between 190° and 213 0 , and is de¬ 
pendent upon the state of the atmosphere. 

The thermometer should be a delicately graduated glass 
tube, made to show the largest possible fraction of a degree 
between those shown in the table. It may be immersed 
in a kettle of steam, but more advantageous results can be 


THERMO ME TRIC LE VELING. 


403 


obtained by using some sort of steam-boiler which will 
bring the larger portion of its surface into immediate con¬ 
tact with a good current of steam. An apparatus of this 
sort may consist of a cylindrical boiler from the center of 
which rises a chimney about 2 inches in diameter by 4 inches 
high, open at the top, and covered by a similar inverted 
chimney, the whole being covered again by a still larger 
chimney; so that the current of steam rising through the 
inner chimney will circulate down through the middle one 
and up through the outer and off through a central vent, 
through which latter the thermometer will be inserted 
through the interior flue. Such a double passageway pre¬ 
vents the condensation of steam on the interior walls. 

Table XXII. 


ALTITUDE BY BOILING-POINT OF WATER. 


Boiling-point. 

Altitude. 

Boiling-point. 

Altitude. 

Degrees (Fahr.). 

Feet. 

Degrees (Fahr.). 

Feet. 

I9O 

I 1,720 

208 

2,050 

195 

8,950 

209 

1.545 

200 

6,250 

2 10 

1,020 

202 

5,185 

2 I I 

510 

204 

4,130 

2 12 

O 

206 

3,085 

213 

- 505 


The lack of delicacy in this instrument is evident when it 
is realized that an error of o. 1 degree in the temperature will 
cause an error of over 80 feet in the determination of eleva¬ 
tions. In addition to being subject to all the errors of meas¬ 
urement by barometer, measurement by thermometer is also 
subject to errors in graduation of the thermometer, lack of 
precision in reading, the quality of glass, and the form of the 
vessel containing the water, as well as the purity of the 
latter, salts in solution materially affecting the boiling-point. 


PART IV. 


OFFICE WORK OF TOPOGRAPHIC MAPPING. 


CHAPTER XIX. 

MAP CONSTRUCTION. 

178. Cartography. —Cartography is the art of construct¬ 
ing maps either (1) from existing material or (2) from original 
surveys. It includes not only the processes of copying, re¬ 
ducing or combining, platting or sketching maps, but also of 
incorporating into them such data as may be obtained from 
text notes or verbal descriptions of the territory represented. 

The expert cartographer must therefore be not only a good 
draftsman, familiar with the methods of map construction and 
the conventional signs commonly employed, but he must be 
possessed of such actual knowledge of map-making as is only 
gained by practical experience in field surveying. Moreover, 
he must be able to distinguish between the quality and value 
of the various map materials which he is to utilize, discerning, 
by his knowledge of topographic forms, the good from the 
bad, and especially that which is based on original surveys 
from that which has been compiled from hearsay or existing 
map sources. 

The draftsman or topographer who makes a map from 
original notes taken in the field is not a cartographer in the 



MAP PROJECTION. 


405 


truest sense of the word. He should know some of those 
details of map construction with which the cartographer is 
familiar, as the projection of the map, conventional signs to be 
employed, and the values of scales, etc. He need not neces¬ 
sarily be familiar, however, with the relative value of existing 
map material, nor be possessed of especial discernment in the 
compilation and utilization of the same. 

179. Map Projection—Having executed the primary 
triangulation (Chap. XXV) and computed the geodetic co¬ 
ordinates of the initial points (Chap. XXIX), these are platted 
on a plane-table sheet by the aid of a projection. This is a 
rectangular diagram on which unit meridians and parallels are 
platted to the scale of the map, and which thus serve as bases 
from which to measure the differential latitudes and longitudes 
of the points so that they may be platted by these co-ordi¬ 
nates, much as the points of a traverse are platted by latitudes 
and departures. (Art. 90.) 

The only absolutely true map is a model of the terrestrial 
globe; but as globes are too awkward for general use, recourse 
is had for purposes of map publication to various forms of 
map projections, which are numerous in variety and are all 
artificial representations upon some plane surface of a sphe¬ 
roidal surface. For surveys extending over a large area it is 
necessary to adopt some method of projection by which the 
convergence of the meridians is shown as on a curved surface, 
and the distances are reduced to sea-level. Where areas which 
are to be mapped are small, the positions of points and the con¬ 
struction of the map may be fixed as upon a plane surface, 
by showing meridians of longitude and parallels of latitude 
as parallel straight lines at right angles to each other. It is 
practically impossible to fix limits within which the first or the 
second of these methods must be employed, as they are not 
only affected by the area covered, but by the scale of the map. 

180. Kinds of Projection. —The varieties of map projec¬ 
tions cannot be more clearly characterized than is done by 


406 


MAP CONSTRUCTION. 


Prof. Dr. Friedrich Umlauff in his admirable little treatise on 
“ The Understanding of Maps,” published in Leipsic in 1889, 
from which the following is freely translated: 

In drawing a small-scale map of a considerable area there 
must be considered: 1, the scale; 2, the projection by which 
it is made; and 3, the manner in which the spheroidal surface 
of the earth as a whole or in part is transferred to the plane 
of the surface. 

A spherical surface cannot be spread on a plane without 
tearing, stretching, or folding; hence maps can never exhibit 
a perfectly true picture of the area represented. Thus there is 
simply a question of selecting a mode of representation which 
shall come as close as possible to the original. To solve this 
problem, various kinds of projections have been devised, aim¬ 
ing to plat the so-called degree-net of the globe, meridians 
and parallels, or a part of it, on a plane surface. There are dis¬ 
tinguished, especially, (1) perspective projections, (2) non-per¬ 
spective projections, (3) conical and (4) cylindrical projections. 

181. Perspective Projections —To project a figure from 
a spherical surface on a plane, nothing occurs to one more 
readily than to employ the same method that is used to depict 
any object in space, as a landscape; namely, by perspective 
drawing. The methods of platting based on the principles of 
the perspective are called perspective projections. The visual 
rays going from the eye to all points of the original are 
imagined to be cut by the plane of the drawing, and the point 
in the picture representing each point in the object is assumed 
to be the point where the visual ray in question cuts the plane 
of the drawing, d he position of this plane is assumed to be 
perpendicular to the ray striking the middle of the area to be 
represented. A difference of the picture can only arise, in 
perspective projections, by a different position of the eye 
with relation to the surface of the sphere. 

If the eye first of all is supposed to be placed at the center 
of the globe, we obtain the gnomonic or central projection. 


PERSPE C TIVE PR OJEC Tip NS. 


407 


As the visual rays pass from the eye through the various lines 
of the degree-net, they are inclined to the plane of drawing at 
smaller and smaller angles the farther they deviate from the 

" 4 . 

center of the plane of drawing, and it is evident that this 
angle must finally dwindle to zero degrees—that the outermost 
visual rays run parallel with the plane of the picture and 
therefore do not intersect-it. 

Thus the circles of the degree- 
net become farther and farther 
apart as we approach the periph¬ 
ery of the map. (Fig. 115, a.) 

If we imagine the eye placed 
at an infinite distance from the 
globe, we obtain the orthographic 
or parallel projection , so called 
because all the rays coming from 
the eye appear parallel and 
therefore strike the plane of drawing at right angles (Fig. 11 5, £). 
The parallel projection permits the representation of a com¬ 
plete hemisphere, which is impossible with the central projec¬ 
tion. 

The third perspective mode of platting is the stereographic 
projection , in which the eye is supposed to be placed at the 
surface of the sphere itself. Here, too, the visual rays diverge 
more and more toward the edges of the picture, but they 
intersect it at greater angles than in the central projection, 
and even the outermost ray still strikes the plane of the 
picture, so that this projection, too, permits the representation 
of a complete hemisphere. (Fig. 116, a.) 

Finally, if the eye is placed outside of the sphere, but at 
a little distance, we obtain th external projection (Fig. 116, b), 
which, however, is but very rarely used. 

To obtain an idea of the networks produced by these 
perspective projections one has to take other things into con¬ 
sideration. If the eye-point, aside from its distance from the 



Fig. 115.—Gnomonic ( a ) and 
Orthographic ( t >) Projections. 
















408 


MAP CONSTRUCTION. 


terrestrial globe, lies in the axis of revolution of the earth, the 
projection is called polar; if the eye-point lies in the plane of 
the equator, the projection is called equatorial; if the eye-point 



Fig. 116.—Stereographic ( a ) and External ( b ) Projections. 


lies outside the plane of the equator and outside the earth’s 
axis of revolution, the projection is called horizontal. Thus, 



Fig. i 17.—Orthographic Equa¬ 
torial Projection. 


Fig. 118.—Orthographic Horizon¬ 
tal Projection. 


disregarding the external projection, we obtain the following 
nine kinds of perspective projections: 

1. Orthographic polar, equatorial, and horizontal projec¬ 
tions. (Figs. 115, 117, and 118.) 































PEKSPECTIVE PROJECPIONS. 


409 


2. Central (gnomonic) polar, equatorial, and horizontal 
projections. (Fig. 115.) 



Fig. 119.—Stereographic Equato¬ 
rial Projection. 


Fig. 120.—Stereographic Meridio¬ 
nal Projection. 


3. Stereographic equatorial, meridional, and horizontal pro¬ 
jections. (Figs. 119, 120, and 121.) 



Fig. 121.—Stereographic Horizon¬ 
tal Projection. 


Fig. 122.—Lambert’s Surface-true 
Central Projection. 


Not all of these modes 
plication. 


of map-platting find practical ap- 



























































MAP CONSTRUCTION. 



\p Maps must comply with certain requirements: 

1. They must be angle-true or conformable ; that is to say, 
parallels and meridians must intersect on the map at the same 
angles as on the original. 

2. They must be surface-true or equivalent; that is to say, the 
areas of given tracts on the original and on the map must agree. 

From the standpoint of practical cartography surface equiv¬ 
alence is most important, because geographic comparisons 
relate mostly to phenomena manifesting their unformity or 
diversity over areally extended regions. From this last-named 
requirement arose especially Lambert’s surface-true central 
projection , which departs from the perspective modes of plat¬ 
ting. It received its name from the fact that at all points of 
equal zenith distance from the middle of the area represented 
the distortions are the same. The equator and the central 
meridian appear as two straight lines perpendicular to each 
other; the other meridians appear as circles, the parallels as 
elliptic curves. (Fig. 122.) Lambert’s surface-true central projec¬ 
tion is not a perspective projection; neither is the so-called globu¬ 
lar projection, invented by the Sicilian Nicolosi, the distinguish¬ 
ing feature of which is that all meridians and parallels are equally 
divided. This is used especially as an equatorial projection. 

Finally, there is a special mode of representation of the whole 
surface of the earth, related to the perspective projections, and the 
origin of which dates back to Ptolemy: the star projection. Every 
polar projection of the northern hemisphere maybe extended into 
a representation of the whole surface of the earth, by appendages 
or wings; the southern half of the earth then divides into four or 
eight parts, to which is given the form of spherical triangles or 
star-like protruberances. The dividing meridians are so chosen 
as to avoid any cutting up of land masses as much as possible. 
For this reason such a star-polar projection is not suitable for 
representing the oceans. Dr. Jager has devised an eight-rayed 
star projection, which was improved by Dr. Petermann; 
H. Berghaus has drawn a similar one with five appendages. 


C }’LINDER PROJECT10NS. 


41 I 

182. Cylinder Projections. —If we imagine the surface of 
the earth circumscribed by a cone tangent to it along a par¬ 
allel, we obtain a conical projection; if the surface of the earth 
appears replaced by a cylinder tangent to it at the equator, we 
obtain the cylindrical or Mercator projection. 




Fig. 124.— Enu 1 distant Flat Pro¬ 
jection. 



Fig. 125.—Mercator’s Cylinder 
Projection. 


If we imagine the equator as the middle parallel, quite a 
broad zone of the globe north and south of the equator may 
be considered as coinciding with the surface of the cylinder. 
On this cylinder the meridians are represented as straight lines, 
















































































































412 


MAP CONSTRUCTION. 


and the equator and parallels as circles of equal length cutting 
the parallels at right angles (Fig. 123, a). To represent a zone 
at a higher latitude, we imagine, instead of the tangent cylinder, 
an intersecting one, also cutting the earth’s surface in the middle 
of the area to be represented. (Fig. 123, b.) If thereupon 



Fig. 126.—Van der Grinten’s Circular Projection. 


we cut the cylinder along a meridian, we obtain two systems 
of straight lines intersecting at right angles, representing the 
parallels and meridians. Maps on such projections are in 
general called flat maps. If the distances of the various parallels 
from each other and also of the meridians are all equal, we obtain 
a network of square meshes, as shown in equidistant flat maps. 


























































































C YLIND ER FROEJECTIONS. 


413 


(Fig. 124.) On such maps the distortion of the surfaces increases 
greatly as we approach the pole, because the parallels, instead 
of dwindling to zero, preserve the same length in all latitudes, 
while the meridians retain the natural length. This incon¬ 
venience is avoided in Mercator's projection by increasing the 
distances between the parallels toward the two poles at the same 
ratio that the parallels increase compared to the equator. (Fig. 
125.) Mercator’s projection is well adapted to maps rcpresent- 



Fig. 127.— Babinet’s Homolographic Projection of the Whole Sphere. 

ing the distribution of general, especially physical, conditions 
over the whole surface of the earth, and for sea-charts, as any 
direction may be represented upon it by a straight line. 

A modification of the cylinder projection is found in the 
Sanson-Flamsteed projection. According to this the parallels are 
drawn as parallel equidistant straight lines, and on these, to the 
right and left of the middle meridian, the degrees of longitude 
are marked in their true size, and the corresponding points of 
intersection are connected by curves representing the meridians. 
If the equator be drawn as a straight line and the central meridian 
also as a straight line of half the length of the equator, we obtain 
an elliptic picture of the whole surface of the globe according 
to Mollweide’s or Babinet’s homolographic projection. (Fig. 127.) 

























414 


MAP CONSTRUCTION . 


Van der Grinten has recently devised a homolographic projec¬ 
tion, not unlike Babinet’s, for the representation of the whole 
world within a circular bounding meridian. The number of 
meridians shown is double the number of parallels, so as to present 
the meridians of the whole world in one plane of projection. 
This projection occupies a middle ground between Mercator’s 
nautical chart of the world and Mollweide’s homolographic map 
of the world, avoiding the distortions of the former and the rapid 
departure from true angles of intersection in the latter. It 
furnishes a natural and suggestive method of showing the whole 
world upon a single projection without violent departure from 
the true areas and shapes of the features shown. 

183. Conical Projections. —Conical projections are quite 
analogous to cylinder projections. A certain zone of the globe 
which is to be represented is conceived to be replaced by a zone 
on the surjace oj a normal cone , either tangent to the sphere or 
intersecting it. (Figs. 128 and 129.) The parallels are drawn 
c 




Fig. 128.—Tangent Cone 
Projection. 


Fig. 129. — Intersecting 
Cone Projection. 


on the surface of the cone as parallel conical circles, while the 
meridians are drawn as straight lines on the conical surface. 
If the surface of the cone is developed, the parallel circles appear 
as arcs of concentric circles whose common center is the apex 
of the cone, while the meridians appear as straight lines con- 
















CONICAL PROJECTIONS. 


415 


verging to that center. The most important conical projections 
are those of Mercator, Lambert, and Bonne. 

An ordinary or equidistant conical projection based on a 
tangent cone shows the meridians as straight lines proceeding 


c 



Fig. 130.—Equal-spaced Conical Projection. 


c 


50 60 TO 



Projection. 



Projection. 


from the apex of the cone at equal angles, while the parallel 
circles are equal-spaced circular arcs with the same apex as 
center. (Fig. 130.) In Mercator's conical projection the dis¬ 
tortion is diminished by making the cone pass through two par¬ 
allels of the area to be represented, so that two parallels of the 
sphere, instead of one, coincide with their pictures. (Fig. 131.) 
This is the projection on which the maps of our common atlases 







































4 i 6 


MA r CONS I R UC1 ION . 


and geographies are drawn. Lambert’s equivalent conical pro¬ 
jection is based on an intersecting 
cone, and the distances of the 



parallels increase with increase 
of latitude at such rate that the 
meshes included by them and the 
meridians show the same areas 
as on the sphere. (Fig. 132.) 


Bonne’s projection is a pro¬ 
jection on the tangent cone in 


Fig. 133. —Bonne’s Projection. 


the center of the map, the parallel curves being drawn in the 
same way as in the ordinary conical projection. On these par¬ 
allel curves, on both sides of the meridian, the parallel degrees 
are marked in their true size, and the points of intersection are 
joined by steady curves which give the meridians. (Fig. 133.) 

184. Constructing a Polyconic Projection.—The polyconic 
projections is that best suited to accurate topographic or geo¬ 
graphic mapping as it corresponds most nearly to the spheroidal 
shape of the earth. It is the projection of a series of cones parallel 
to each parallel of latitude to be drawn on the map. Assume 
the scale of the map as one mile to one inch, or 1:63,360. For 
this scale it will be sufficient to draw the meridian and latitude 
lines at intervals of every five minutes or approximately five 
inches apart, though single minute lines may be drawn if desired. 
The construction of such a projection, requires great care and 
accuracy in drafting. The process is as follows: 

Rule a fine straight, vertical line down the center of the sheet. 
(Fig. 134.) On this lay off the lengths of the several five- 
minute spaces in latitude, these being the dV s as taken from 
Table XXIII for the scale 1:63,360. This fixes the points 
of intersection of the parallels at every five minutes with the 
central meridian. Erect perpendiculars on each of these 
points, and draw these across the map at right angles to the 
central meridian, as shown in dotted lines. On these lav 
off the quantities dm (Table XXIII) for half the distance of 










CONSTRUCTING A TOLY CO NIC PROJECTION . 417 

five minutes, that is, for 2' 30" and 7' 30" on either side of 
the central meridian and corresponding to the latitude as 
obtained fiom the table. On the points so obtained on each 



Fig. 134.— Construction of Polyconic Projection. 

30' of latitude and longitude. Scale 2 miles to 1 inch. Construction lines 

dotted. Final projection lines full. 

approximate parallel erect short perpendiculars, and on these 
lay off the small quantity dp corresponding to the dm , and 
connect the various dp's by strai gin n nes in a horizontal and 
vertical direction. The result will be a projection similar to 
that shown in full lines in the figure. 

























418 


MAP CONSTRUCTION . 


185. Projection of Maps upon a Polyconic Develop¬ 
ment. —The following table (Table XXIII) is arranged for 
the projection of maps upon a polyconic development of the 
Clarke spheroid. It is on a scale of one mile to one inch, and 
is computed from the equator to the pole, the unit scale being 
>ne selected for presentation here, as that most generally use- 
mi, since the quantities shown in the table can be most readily 
reduced to those applicable to other scales which are even 
multiples of one mile to one inch. They are reproduced 
from the Smithsonian Miscellaneous Tables, for which they 
were prepared by Prof. R. S. Woodward. 

The following formulas are those used in the preparation 
of this and similar tables, and are derived from the United 
States Coast and Geodetic Survey Report for 1884: 

For lengths of degrees of the meridian (dm) and parallel (dp) we have 
dm =: hi 132'".09 — 566 m .05 cos 2<p -f- i ,n .20 cos 4 (p — o'".003 cos 60 ; 
dp = in 4i5 m .io cos 0 — 94 m .54 cos 30 -j- o m .i2 cos 50 , neglecting 
smaller terms, 
where 0 = the latitude. 

We have also the square of the eccentricity, 

d% __ 

P = o 006768658 = --—. 


a 

N = -:— . 9 = normal produced to minor axis; .... (33) 

(1 — e l sin 2 0)1 ’ 

1 — 

Rm = N z - - — = radius of curvature in the meridian; . . . (34) 

Rp = N cos 0 = radius of the parallel;.(35) 

r — N cot 0 = radius of the developed parallel or side of 

the tangent cone;.(36) 

0 = n sin 0, 

in which n is any arc of the parallel to be de¬ 
veloped, and 0 the angle which it subtends at the 
vertex of the cone when developed. (37) 

For projecting from the middle meridian the points of intersection of 
the meridians and parallels we have, using rectangular coordinates X 
and Y, 

x ~ r sin 0.(38) 

and 

Y — 2r sin 2 10 .(29) 










COORDINATES FOR PROJECTION OF MAPS . 


419 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale 33^0* or one i nc h to one mile. 

(From Smithsonian Tables.) 


* 

Latitude of 
Parallel. 

Meridional Dis- 
^ tances from 
Even-degree 
Parallels. 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 

ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 

5 ' 

long. 

1 o' 

long. 

15 ' 

long. 

20 ' 

long. 

25' 

long. 

30 ' 

long. 


inches. 

inches. 

inches 

inches. 

inches. 

inches. 

inches. 

cu 

-o at 
p > 



o°oo' 


5-764 

II .529 

17.293 

23.058 

28.822 

34.586 

■*-> t- 

c - 
c 

o° 

1° 

IO 

II .451 

s. 764 11. 528 

17.293 

23-057 

28.821 

2 a. 5 S 5 

*— 



20 

22.901 

5.76411 528 

17.292 

23.056 

28.821 34.585 


inches. 

inches. 

30 

34-352 

5.764 11.528 

17.292 

23.056 

28.820 34.583 

5 ' 



40 

45•803 

5-764 

II.528 

T7.291 

23-055 

2S - 819 1 34.583 

0.000 

0.000 

50 

57-254 

5-764 

II.527 

17.291 

23.054 

28.818 

34.582 

IO 

.OOO 

.OOO 









15 

.OOO 

.OOI 

I OO 

68.704 

5.764 

II.527 

17.29I 

23.054 

28.818 

34 - 58 i 

20 

.OOO 

.OOI 









25 

.OOO 

.002 

IO 

11.451 

5.763 II. 526 

17.289 

23.C52 

28.816 

34-579 

30 

.OCO 

.003 

20 

22.901 

5-763 

11-525 

17.28S 

23.050 

28.813 34.576 




30 

34-352 

5.762 

II -524 

17.287 

23 019 

28.811 

34-573 




40 

45.803 

5.762 

II.524 

17-285 

23 047 

28.809 34.571 




50 

57-254 

5-761 

II.523 

17.284 

23-015 

28.807 34-568 




' 2 OO 

68.704 

5-761 

II .522 

17.283 

23.044 

vn 

O 

00 

CO 

34-565 


2° 

3 ° 

IO 

ii- 45 i 

5.760 

II.520 

17.281 

23.041 

28.8or 

34 .561 

5 

0.000 

0.000 

20 

22.902 

5-759 

II. 519 

17.278 

23.038 

28.797 34-556 

10 

.001 

.OOll 

30 

34-353 

5-759 

11 - 5 1 7 

17.276 

23.035 

28.794 34-552 

15 

.001 

.002 

40 

45-804 

5-758 

11.516 

17.274 

23.032 

28.790 34 . 54 S 

20 

. 002 

.003 

50 

57-254 

5-757 

11.514 

17.272 

23.029 

28.786 34. 543 

25 

.004 

.005 









30 

.005 

.008 

3 00 

68.705 

5.756 

II-5I3 

17.270 

23.026 

28.783 34-539 

| 




IO 

11.451 

5-756 

11 5 11 

17.267 

23.022 

28.778 34.533 




20 

22.902 

5-754 

11.509 

17.264 

23 018 

28.773 34-527 




30 

34-353 

5-753 

11.507 

17.260 

23.014 

28.767 34.520 




40 

45.804 

5-752 

11 - 505 

17 • 257 

23 • 010 

23 .702 34.514 




50 

57-255 

5-751 

11-503 

17-254 

23.006 

28.757 34-508 


4 ° 

5 ° 

4 00 

68 . 706 

5-750 

11.501 

I7-25T 

23 • 002 


04- 




10 

11.451 

5-749 

11.493 

17.247 

22.996 

28.746 34-495 

5 

10 

0.000 

. 001 

0.000 

. OOI 

20 

22.903 

5-748 

11.496 

17-243 

22.991 

28.739 34-487 

15 

003 

.003 

30 

34-354 

5.746 

11.493 17.240 

22.986 

28.733 34-479 

20 

.005 

71 
. 006 

40 

45.805 

5-745 

11.490,17.236 

22.981 

28.726 34.471 

25 

.007 

.009 

50 

57-256 

5-744 

11.4S8 17.232 

j 

22.976 

28.720 34.463 

30 

.011 

.013 

5 00 

68.708 

5-743 

11.485^17.228 

22.970 

28.713 34-456 
1 






























































































420 


MAP CONSTRUCTION. 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 


Scale or one inch to one mile. 



«> <u 

q a a 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 





~ 2 A 








ORDINATES OF 

O _! 
vJi 

5 ^ A 








DEVELOPED 


.2 w 

T -5 O (U ^ 







/ 

PARALLEL. 


E > # 


5 ' 

10 

15 ' 

20 


30 


dp 


hJ 

2 

dl 

long. 

long. 

long. 

long. 

long. 

long 





inches. 

inches. 

inches. 

inches. 

inches 

inches. 

inches. 

•0 ^ 

p > 



5 °oo' 

68.708 

5 

•743 

H.485 

17.228 

22.970 

28.713 

34-456 

C C 

O ■ 

5 ° 

6° 

IO 

II.452 

5 

741 

II.482 

17 . 22 -I 

22.964 28.705 34.446 

►J 



20 

22.903 

5 

739 

II.479 17.218 

22.958 28.697 34.436 


inches 

inches. 

30 

34-355 

5 

738 

II.476 17.213 

22.951 

28.689 34.427 




40 

45 - 806 

5 

736 

II.472 17.209 22.945 28.681 

34-417 

5 ' 

0.000 

O. OOO 

50 

57.25S 

5 

735 

II .469 17.204 22.938 28.673 34.408 

10 

• OOI 

.002 










15 

•003 

004 

6 oo 

68.710 

5 

733 

II.466 17.199 

22.932 28 665 34-398 

20 

.006 

.007 










25 

.009 

.OII 

IO 

11.452 

5 

73 i 

II.462 17.193 

22.924 28.656 34.387 

30 

013 

.016 

20 

22.904 

5 

729 

II.458 17.188 

22.917 28.646 

34-375 




30 

34-356 

5 

727 

II.455 17.182 

22.910 28.637 

34-364 




40 

45.808 

5 

726 

II.451 

17.177 

22.902 

28.628 

34-353 




50 

57.260 

5 

724 

II.447 

17.17] 

22.894 28.618 

34-342 




7 oo 

68.712 

5 

722 

H -443 

17.165 

22.8S7 28.609 

34-330 


7 ° 

8’ 

IO 

11.452 

5 

720 

u -439 

17.159 

22.878 

28.598 

34-317 

5 

0.000 

0.001 

20 

22.905 

5 

7 i 7 

11 •435 

17.152 

22.869 

28.587 

34-304 

10 

.002 

.002 

30 

34-358 

5 

715 

11.430 

17.146 

22.861 

28.576 

34-291 

15 

• 005 

.005 

40 

45 -8 io 

5 

7 i 3 

11.426 

17-139 

22.852 

28.565 

34-278 

20 

.008 

.009 

50 

57.262 

5 

711 

11.422 

17.132 

22.843 

28.554 

34-265 

25 

• 013 

.014 










3 ° 

.018 

.021 

8 oo 

68.715 

5 

709 

11.417 

17.126 

22.834 

28.543 

34.252 




IO 

n -453 

5 

706 

11.412 

17.II9 

22.825 

28.531 

34-237 




20 

22.906 

5 

704 

11.407 

17.Ill 

22.815 

28.519 

34.222 




3o 

34-359 

5 

701 

11.403 

17.IO4 

22.805 

28.507 

34.208 




■40 

45-812 

5 

699 

11.398 

I7.096 

22.795 

28 494 

34-193 




50 

57-265 

5 

696 

H -393 

17.089 

22.786 

28.482 

34-178 


9° 

IO° 

9 00 

68.718 

5 

694 

n.388 

17.082 

22.776 

28.470 

34-163 



— 










5 

0.001 

0.001 

10 

n -454 

5 

691 

11.382 

17.073 

22.764 

28.456 

34-147 

10 

• 003 

• 003 

20 

22.907 

5 

688 

n -377 

17-065 

22.754 

28.442 

34-130 

15 

.006 

.006 

30 

33-36 i 

5 

686 

H- 37 L 

17-057 

22.742 

28.428 

34-114 

20 

.010 

.Oil 

40 

45-814 

5 

683 

11.366 

17.O49 

22.732 

28.415 

34-097 

25 

.016 

.018 

50 

57-268 

5 

680 

11.360^17.040 

22.720 

28.401 

34.081 

30 

.023 

.026 

10 00 

68.722 

5.677 

ii *355 1 7 *032 

22.71028 387 

1 

34-064 












































































COORD IN A IES FOR PROJECTION OF MAPS. 


421 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale or one inch to one mile. 


Latitude of 
Parallel. 

Meridional Dis- 
^ tances from 
^ Even degree 
Parallels. 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 

ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 

5 ' 

long. 

10' 

long. 

15 ' 

long. 

20' 

long. 

25 ' 

long. 

3 o' 

long. 


inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

•u rt 
a > 



io°oo' 

68.722 

5-677 

u -355 

17.032 22.710 

1 

28.387 

34-064 

•as 

C G 

0 ’ 

IO° 

11° 

IO 

II.454 

5.674 

11 •349 

17.023 22.698 28.372 34.046 

►J 



20 

22.909 

5-671 

H -343 

17.014 22.6S5 28.357 34.028 


inches. 

inches. 

30 

34.263 

5.668 

n -337 

17.005 22.673 28.342 34.OIO 




40 

45-817 

5-665 

II- 33 I 

16.996 

22.66l 28.327 33.992 

5' 

0.001 

0.001 

50 

57-272 

5.662 

11.324 

16.987 

22 649 28.311 33.973 

IO 

.003 

• 003 









15 

.006 

.007 

II OO 

6S.726 

5-659 

ii.318 

16.978 

22.637 28.296 33-955 

20 

.Oil 

■013 









25 

.018 

.020 

IO 

H -455 

5-656 

n.312 

16.968 

22.624 28.280 33.935 

30 

.026 

.028 

20 

22.910 

5.652 

n.305 

16.958 

22.610 28.263 33 - 9 I 5 




30 

34-365 

5-649 

11.298 

16.948 

22-597 28.246 33.895 




40 

45.820 

5-646 

11.292 

16.938 22 584 28 230 33.875 




50 

57-275 

5.642 

11.285 

l6 928 

22.570 28.213 33-855 




12 OO 

68.730 

5-639 

11 278 

16.918 

22.557 28.196 33.835 


12° 

I 3 C 

IO 

IT .456 

5-636 

11.271 

16.907 

22.542 28.178 

33-814 

5 

0.001 

0.001 

20 

22 912 

5-632 

11.264 

16.896 22 528 28.l6o 

33-792 

10 

.003 

.004 

30 

34-367 

5.628 

n.257 

16.885 22.514 28.I42 

33-770 

15 

.008 

.008 

40 

45-823 

5.625 

11.250 

16.874 22.499 28.124 

33-749 

20 

.014 

.015 

50 

57-279 

5.621 

11.242 

16.864 

22.485 28.I06 

33-727 

25 

.021 

.023 









30 

.031 

-033 

13 OO 

68.735 

5 • 618 

11.235 

16.853 

22.470 

28.088 

33•706 




IO 

n -457 

5.614 

11.227 

16.841 

22.455 

28.069 

33.682 




20 

22.913 

5.610 

11.220 

16.829 

22.439 

2S.049 

33-659 




30 

34-370 

5.606 

11.212 

I6.8l8 

22.424 

28.030 

33-635 




40 

45-827 

5.602 

11.204 

16.806 

22.408 

28.010 

33.612 




50 

57-284 

5-598 

11.196 

16.794 

22.392 

27- 99 1 

33-589 


14 ° 

15 ° 

14 OO 

68.740 

5-594 

11.188 

16.783 

22.377 

27.971 

33-565 












5 

0.001 

0.001 

IO 

11.458 

5.590 11.180 

16.770 

22.360 

27.950 

33-540 

10 

.004 

.004 

20 

22.915 

5.58611 172 

16.758 22.344 

27.930 

33-515 

15 

.009 

.009 

30 

34-373 

5-582 

11.163 

16.745 22.327 

27-909 33-490 

20 

• Ol6 

.017 

40 

45-830 

5.578 

1 I.I 55 

16.733 22.310 

27.888 33.465 

25 

-025 

.026 

50 

57.288 

5-573 

11.147 

16.720 22.294 

27.867 33.440 

30 

•035 

.038 

15 OO 

l- 

68.746 

5.569 

n • 138 

16.708 22.277 

27.846 33.415 

1 


















































































422 


MA P CONS TR UCT10N* 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale 635S0* or one inch to one mile. 



S2 v 

b e a 

ABSCISSAS OK 

DEVELOPED PARALLEL, dm 




V*-. 

— £ 







ORDINATES OF 

o 

0J V 

= xVA 







DEVELOPED 

— 

p 2 

W P'- 1 

5 ' 


15 ' 



/ 

PARALLEL. 

.- .U 

rt ^ 

■C 5 > X 

V 2 M p_ 

10 

20 

25 ' 

30 


dp 



2 

dl 

long. 

long. 

long. 

long. 

long. 

long. 





inches. 

inches. 

I 

inches. 

inches. 

inches. 

inches. 

1 

inches. 

-a td 

3 > 



i5°oo' 

68.746 

5-'569 

II.138 

16.708 

22.277 

27.846 33-415 

4-i U 

• — <i> 

U w 
c c 

C 

15 ° 

16 0 

IO 

U -459 

22.917 

5 • 565 11 • 130 
5.560 11.121 

16.694 

16.681 

22.259 

22.241 

27.824 A'l . -ISO 




20 

27.802 

33-362 


inches. 

inches. 

30 

34-376 

5.556 11.112 

16.667 

22.223 

27-779 33-335 

5 ' 



40 

45-834 

5-551 

II. 103 

16.654 

22.206 

27-757 33 308 

0.001 

0.001 

50 

57-293 

5.547 11-094 

16.641 

22.l88 

27-735 33-282 

10 

.004 

.004 









15 

.009 

.OIO 

16 oo 

68.752 

5.542 11.085 

16.628 

22.170 

27.713 33-255 

20 

.017 

.0181 









25 

30 

. 026 

. 028 

IO 

11.460 

5-538 

II .076 

16.613 

22.151 

27.689 

33.227 

.038 

.040 

20 

22.919 

5-533 

II .066 

16.599 

22.I32 

27-665 33.198 




30 

34 379 

5.528 11.057 

16.585 

22.II3 

27.642 33-170 




40 

45-838 

5.524 11.047 

16.571 

22.094 

27.618 

33 142 




50 

57.298 

5-519 

II.038 

16 556 

22.075 

27-594 

33 -H 3 







17 oo 

68.758 

5.514 11.028 

16.542 

22.056 

27-571 

33 085 


17 ° 

18 0 

10 

11.^61 

5-509 

II .018 

16.527 

22.036 

27-546 

33 055 

5 

0.001 

0.00T 

20 

22.921 

5-504 

II .008 

16.512 

22.016 

27.521 

33 025 

10 

.005 

.005 

30 

34-382 

5-499 

IO.998 

16.497 

21.996 

27-495 

32.994 

15 

.011 

.011 

40 

45.843 

5-494 

IO.988 

16.482 

21.976 

27.470 

32 964 

20 

.019 

.020, 

50 

57.304 

5-489 

IO.978 

16.467 

21.956 

27-445 

32 934 

25 

.029 

• 031 

18 00 

68.764 







30 

.042 

•044 

5-484 

IO.968 

16.452 

21.936 

27.420 

32.904 



10 

20 

30 

40 

11.462 
22.924 
34-386 
45.848 

5-479 

5-473 

5.468 

5-463 

ro -957 
10 947 
10.936 
10.926 

16.436 

16.420 

16.404 

16.3S9 

21.915 
21.894 
21.872 
21.852 

27-394 

27.367 

27-341 

27 - 3 I 5 

32.872 
32.840 
32.809 
32.777 







50 

19 00 

57 - 3 io 

68.771 

5-458 

5-452 

10.915 

10.905 

16.373 

16.357 

21.83O 

21.809 

27.288 

27.262 

32.746 

32 - 7 I 4 


19 ° 

20° 

5 

0.001 



n.463 







0.001 

10 

5-447 

10.893 

16.340 

21.787 

27.234 

32.680 

10 

• 005, 

.005 

20 

22.926 

5-441 

10.882 

16.324 

21.765 

27.206 

32.647 

15 

.012 

.012 

30 

34-390 

5-436 

10.871 

16.307 

21.742 

27.178 

32.614 

20 

.021 

.022 

40 

45.853 

5-430 

10.860 

16.290 

21.720 

27 150 

32.580 

25 

.032 

.034 

50 

57-316 

5-424 

10.849 

16.274 

21.698 

27.I 23 ! 

32-547 

30 

.046 

.049 

20 00 

68.779 

5419 

10.838 

16.257 

21.676 

27.095' 

32.513I 








1 


1 

1 



J 



































































































COORDINATES FOR PROJECITON OF MAPS, 


423 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale or one inch to one mile. 


ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 




•0 a 



3 > 




20° 

21° 

5 c 



hJ 




inches. 

inches. 

5 ' 

0.001 

0.001 

10 

.005 

.006 

15 

.012 

.013 

20 

.022 

.022 

25 

.034 

.035 

30 

.049 

.051 


22° 

23 ° 

5 

0.001 

0.001 

10 

.006 

.006 

15 

.013 

.014 

20 

.023 

. 024 

25 

.036 

•038 

30 

.052 

• 054 


24 ° 

25 ° 

5 

0.002 

0.002 

10 

.006 

.006 

15 

.014 

.014 

20 

•025 

.026 

25 

•039 

.040 

30 

.056 

.058 


Latitude of 
Parallel. 

Meridional Dis 
5 ^ tances from 
^ Even-degree 
Parallel. 


inches. 

20°00' 

68.779 

IO 

II .464 

20 

22.929 

30 

34-394 

40 

45.858 

50 

57-322 

21 OO 

68.787 

IO 

11.466 

20 

22.932 

30 

34-397 

40 

45-863 

50 

57.329 

22 OO 

68.795 

IO 

1i.467 

20 

22.934 

30 

34.401 

40 

45.868 

50 

57-336 

23 OO 

68.803 

IO 

11.469 

20 

22.937 

30 

34.406 

40 

45.874 

50 

57-343 

24 OO 

6S.812 

IO 

11.470 

20 

22.940 

30 

34-410 

40 

45.880 

50 

57-350 

25 OO 

68.821 


ABSCISSAS OF DEVELOPED PARALLEL, dm 


5 

long. 


inches. 

5-419 


413 

407 

401 

396 

390 

334 

378 

372 

366 

359 

353 

347 

34 i 

334 

328 

322 

315 

309 

302 

296 

289 

282 

276 

269 


10' 

long. 


15 ' 

long. 


inches, inches. 

I 

IO. 838^16.257 

IO.826 16.239 
IO.814 16.222 
IO.803 16.204 
IO.791 16.187 
10.779 16.169 

10.768 16.151 


10-755 16.133 
IO.743 16.II5 
10.731 16.097,21.462 
10.719 16.078 21.438 
21.413 


20' 

long. 

inches. 

21 

.676 

21 

•652 

21 

. 629 

21 

.605 

21 

.582 

21 

•558 

21 

•535 

21 

.511 

21 

.486 


25 

long. 


30 ' 

long. 


IO. 707 16.060 
IO.694 16.042 
10 682 16.022 

IO.669 16.003 
IO.656 15.984 
IO.643 15.965 
IO.63I 15.946 

I 

I0.6l8:l5.927 

10.604 15 - 9°7 
10.59P15.887 
10.578 15.867 
10.565 15-847 
10.551 15-827 

15-807 


10.538 


26310.526 


256 

249 

242 

235 


5.227 


10.512 

10.498 

10.483 

10.469 

io .455 


15.789 

I 5.767 

I 5.746 

15-725 

I 5-704 

15 6S2 


21.389 


inches | inches. 

I 

27.095 32.513 

27.065I32.478 
27.036132.443 
27.007 32.408 


26.978 

26.948 

26.919 

26.889 
26.858 
26.828 
26.797 
26.767 

26.736 


21.363 26.704 
21.338 26.672 
21.31226.64131 969 


32.373 

32.338 

32.303 

32.266 
32.230 
32.193 
32.156 
32.120 

32.083 

32.045 

32.006 


21.287 

2I.26l 

21.236 

21.209 
21.182 
21.156 
21.129 
21.102 

21.076 


26.609 

26.577 

26.545 

26 511 
26.47S 
26.445 
26.412 
26.378 


26.345 


21.052 26. 
21.023 26. 
2O.995 26. 
20.967 26. 
20.938 26. 


3L5 


31.930 

31.892 

3I-853 

3 I. 8 I 3 

31-774 

31-733 

3r.694 

3I.654 

31.614 

31-577 


279 31.535 
244 31-493 
209 31.450 
173 3 I .408 


2O.9IO 26. 137 3I.365 



































































424 


MA P COA T S TP UCT10N . 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale 6356a, or one inch to one mile. 



C/5 v 

ABSCISSAS OF 

DEVELOPED PARALLEL, dvi 




Vtx 

Q - 

— u ^ 







ORDINATES 

OF 

o _• 
u a; 

ctj '■*— • — 

e *•?« 







DEVELOPED 


•2 5 = « 







PARALLEL. 

rt ^ 

S S WO- 

5 

10 

15 

20 

25 

3o' 


dp 


w 

*—« 

" dl 

long. 

long. 

long. 

long. 

long. 

long. 












0 _• 




inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

•a nj 

3 > 



25°oo' 

68.821 

5-227 

10-455 

15.682 

20.9IO 

26.137 

3I-365 

’St 2 
c c 
c »-< 

25° 

26° 

10 

II.472 

5.220 

IO.441 

15.661 

20.881 

26. IOI 

31-322 







20 

22.943 

5.213 

IO.426 15.639 

20.852 

26.065 

31-279 


inches. 

inches. 

30 

34-415 

5.206 

IO.412 

15.618 

20.824 

26.029 

31-235 

c ' 

0.002 

.006 

.014 

O.OC2 

.007 

.015 

z 

40 

45.886 

5-199 

10-397 

15.596120.795 

25-993 

31.192 

J 

10 

15 

50 

57-358 

5-191 

IO.383 

15-575 

20.766 

25-958 

31-149 

26 00 

68.830 

5.184 

IO.369 

15-553 

20.737 

25.922 

31.106 

20 

25 

.026 

.040 

.026 

.041 

10 

H-473 

5-177 

10-354 

I5-53I 

20.708 

25.884 

31.061 

30 

.058 

-059 

20 

22.946 

5.169 

10.339 

15-508 

2O.67S 

25.847 

31.017 




30 

34-419 

5.162 

IO 324 

15-486 

2O.648 

25.810 

30.972 




40 

45.892 

5-154 

IO.309 

15-463 

20.6l8 

25.772 

30.927 




50 

57-365 

5-147 

IO.294 

15-441 

0^ c88 oc -nr 

30.882 





* J JJ 




27 00 

68.838 

5.140 

IO.279 

I5-4I9 

20.558 

25.698 

30.838 


27° 

28° 

10 

n-475 

5-132 

IO. 264 

I5-396 

20.528 

25-659 

30.791 

5 

0.002 

0.002 

20 

22.950 

5.124 

IO.248 

15.373 20.497 

25 621 

30-745 

10 

.007 

.007 

30 

34-424 

5 • 116 

IO.233 

15.349 20.466 

25.582 

30.699 

15 

-015 

.016 

40 

45.899 

5.109 

10.218 

15-326 

20.435 

25-544 

30.653 

20 

.027 

.028 

50 

57-374 

5 -ioi 

10.202 

15-303 

20.404 

25-505 

30.607 

25 

.042 

•043 









30 

.06l 

.063 

28 00 

68.849 

5.093 

10.187 

15.2S0 

20.374 

25.467 

30.560 




10 

20 

30 

40 

11.476 

22.953 

34-430 
45•906 

5-085 

5.077 

5-069 
5-061 

10.171 
10.155 
IO.I39 
10.123 

15-256 

15.232 

15.208 

15.185 

20.342 
2O.3IO 
20.278 
20.246 

25-427 

25-387 

25-347 

25.308 

30.513 

30.465 

30.417 

30.369 







50 

29 00 

57-383 

68.859 

5-054 

5-046 

10.107 

10.091 

15.161 

15-137 

20.214 

20.182 

25.268 

25.228 

30.321 

30.274 


29° 

O 

30 

5 

0.002 

0.002 

















10 

11.478 

5-037 

10.075 

15-112 

20.150 

25.187 

30.224 

10 

.007 

.007 

20 

22.957 

5.029 

10.058 

15.087 20.117 

25-146 

30.175 

15 

.016 

.016 

30 

34-435 

5.021 

10.042 

15.063 20.0S4 

25-105 

30.126 

20 

.028 

.029 

40 

45•9 X 3 

5 013 

ro.025 

15.038 20.051 

25.064 

30.076 

25 

.O44 

-045 

50 

57-391 

5.004 

10.009 

15.013 20.018 

1 

25.022 

30.027 

30 

.064 

.065 

30 00 

68.870 

4.996 

9-993 

1 

14.98919.985:24.981 

29.978 

m 





















































































COORDINATES FOR PROJECTION OF MAPS. 425 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale or one inch to one mile. 


C/3 

Q 


<u 
B u 


' , 2 ^ 


ABSCISSAS OF DEVELOPED PARALLEL, dm 


0 

vJH 

"O -z 

1 - 1 

OS - U — 

-3 H « £ 

S 

dl 

5 ' 

long. 

10' 

long. 

15 ' 

long. 

20' 

long. 

25 ' 

long. 

30 ' 

long. 


inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inolle s 

1 

30°oo' 

68.870 4.996 

9 993 

14.989 

19.985 

24 981 

29.978 

10 

11.480 

4.988 1 

I 

9.976 

r 4-963 

19-95 1 

2 4 939 

29.927 

20 

22.g6o 4 979 

9-959 

14 938 

19.917 24.896 

29.876 

30 

34 - 44 ° 

4-971 

9.942 

14.912 

19.883 24.854 

29 825 

40 

45.920 4.962 

9-925 

14 837 

19.849 

24.812 

29-774 

50 

57.400 

4-954 

9.90S 

14-862 

19 - 8 I 5 

24.769 

29.723 

31 00 

68.880 

4-945 

9 891 

14.836 

19 782 

24.727 

29.672 

10 

11.482 

4-937 

9 873 

14.810 

19-747 

24.683 

29.620 

20 

22.964 

4.928 

9.856 

T 4-784 

19.712 

24.640 

29 568 

30 

3 6446 

4.919 

9-838 

14 758 

19.677 

24 596 

29-515 

40 

15-927 

4.910 

9.821 

14-731 

19.642 

24-552 

29.463 

50 

57 - 4°9 

4.902 

9.804 

14.705 

19.607 

24.509 

29.411 

32 00 

68.891 

4 - 89.3 

9.786 

14.679 

19-572 

24 465 

29-358 

10 

11.484 

4.884 

9.768 

14-652 

I 9-536 

24.420 

29.305 

20 

22.967 

4-875 

9 - 75 ° 

14-625 

19.500 

24.376, 29.251 

30 

34 - 45 t 

4.866 

9.732 

14 598 19-465 

24-331 

29.197 

40 

45-934 

4-857 

9 7 i 4 

14-572 

19.429 

24.286 29.143 

50 

57-418 

4.84S 

9.096 

14-545 

19-393 

24 211 

29.089 

33 00 

68.902 

4-839 

9.679 

14-518 

IQ -357 

24.T96 

29.036 

10 

11.435 

4.830 

9.660 

14 490 19 320 

2 A.I 50 

28.980 

20 

22.971 4.821 

9 642 

14.462 

19.283 

24.IO4 

28.925 

30 

34-456 

4.812 

9.623 

14-435 

19.246 

24-058 

28.870 

40 

45-942 

4.802 

9-605 

14.407 

19.210 

24.012 

28.814 

50 

57-427 

4-793 

9-536 

14-379 

19 1 73 

23.966 

28.759 

34 00 

68 913 

4 784 

9-568 

14-352 

1 19-136 

23.920 

28.704 

10 

11.487 

4-774 

9-549 

14.323 19.098 

23.872 

28.647 

20 

22.975 

4-765 

9-530 

14-295 

19.080 

23.825 

28.590 

30 

34462 

4-755 

9 - 5 H 

14 267 19.022 23.778 

28.533 

40 

45-949 

4.746 

9.492 

14.238 18 9S4 

23-730 

28.476 

50 

57-437 

4-737 

9-473 

14.210 

18.946 23.683 

i 

28.420 

35 00 

68 924 

4.727 

9-454 

14.181 

1 18.908 

23.636,28.363 

1 1 


ORDINATES OF 
DEVELOPED 
PAkALLEL. 
dp 


Longitude 

Interval. 

30 ° 

3 i° 


inches. 

inches. 

5 ' 

0.002 

0.002 

10 

.007 

.007 

15 

.016 

.017 

20 

.029 1 

.030 

25 

-045 

.046 

30 

.065 

.067 


32 ° 

33 ° 

5 

0.002 

0.002 

10 

.007 

.008 

15 

.017 

.017 

20 

.030 

.031 

25 

.047 

.048 

30 

.068 

.069 


34 ° 

35 ° 

5 

0.002 

0.002 

10 

.008 

.008 

T 5 

.017 

.018 

20 

•031 

.031 

25 

.049 

• 049 

30 

.070 

.071 














































































426 


MAP CONSTRUCTION. 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 


Scale S3or one inch to one mile. 


Latitude of 
Parallel. 

Meridional Dis- 
^ tances from 
Even-degree 
Parallels. 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 

ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 

5' 

long. 

10' 

long. 

15' 

long. 

20' 

long. 

25' 

long. 

30 ' 

long. 


inches. 

inches. 

! inches. 

inches. 

inches. 

inches. 

inches. 

V _* 

'U nj 

3 > 



35°oo' 

68.924 

4.727 

9-454 

14.181 

18.908 

23.636 

28.363 

-*-» u 

c c 
c ' 

35 ° 

36 ° 

IO 

11.489 

4.717 

9-435 

14.152 

18.870 

23. =87 

28.305 

►—< 



20 

22.978 

4.708 

9-415 

14 123 

18.83I 

23-539 

28. 246 


inches. 

inches. 

30 

34.468 

4.698 

9-396 

14.094 

iS.792 ( 23.490 

2S.188 




40 

45-957 

4.688 

9-377 

14.065 

18.753 23.442 

28.130 

5' 

0.002 

0.002 

50 

57-446 

4.679 

9-357 

14.036 

18.714 23.393 

28.072 

10 

.008 

.008 









15 

.018 

.018 

36 OO 

68.935 

4.669 

9-338 

14.007 

18.676:23.345 

28.014 

20 

.031 

.032 









25 

•049 

.050 

TO 

11.491 

4.659 

9.318 

13-977 

18.636 

23-295 

27-954 

30 

.071 

.072 

20 

22.983 

4.649 

9.298 

13-947 

18.596 23.245 

27.894 




30 

34-474 

4.639 

9.278 

13 917 

18.556 

23-195 

27.835 




40 

45.965 

4.629 

9.258 

13.887 

18.517 

23.146 

27 775 




50 

57 457 

4.619 

9-238 

13-858 

18.477 

23.096 

27 615 




37 00 

68.94S 

4.609 

9.219 

13.828 

18.437 

23.046 

27.656 


37 ° 

38 s 

10 

n -493 

4-599 

9.198 

13-797 

18.396 

22.995 

27-594 




20 

22.986 

4539 

9.178 

13.767 

18.356 

22 944 

27 533 

5 

0.002 

0 002 

30 

34.480 

4-579 

9 - 157 - 

I 3-736 

18.315 

22.S94 

27.472 

10 

.008 

.008 

40 

45-973 

4.568 

9-137 

13.706 

18.274 

22.843 

27.411 

15 

.018 

.018 

50 

57.466 

4-553 

9 -H 7 

I 3-675 

1S.234 

22 792 

27.350 

20 

.032 

033 









25 

.050 

.051 

38 OO 

63 .959 

4 548 

9.096 

13-645 

18.193 

22.741 

27.289 

30 

•073 

.073 

IO 

1 1-495 

4 538 

9 076 

13-613 

18.151 

22.689 

27.227 




20 

22.990 

4-527 

9-055 

13-582 

18.109 

22.637 

27.164 




30 

3 4 4 8 5 

4 5i7 

9-034 

I 3 - 55 I 

18.068* 

22.585, 

27.102 




40 

45.980 

4.506 

9 013 

13-520 

18.026 

22.533 

27.039 




50 

57-475 

4.496 

S 992 

13.488 

17.984 

22.481 

26.977 


39 ° 

1 

40 ° 

39 °o 

00.970 

4.486 

8.971 

13-457 

17-943 

22.429 

26 914 












5 

0.002 

0.002 

10 

ri.497 

4-475 

8.950 

I 3-425 

17.900 

22.375 

26.85 1 

10 

.008 

.008 

20 

22.994 

4.464 

8.929 

13-393 

17.8 = 8 

22.322 

26.787 

15 

.018 

.019 

30 

34-491 

4-454 

8.908 

13-361 

17.815 

22.269 

26.723 

20 

•033 

■033 

40 

45 . 98 S 

4 443 ! 

8.886 

13-330 

17-773 

22.216 

26.659 

25 

.051 

.052 

50 

57-485 

4-433 

8.865 

13.298 

17.730 22.163 

26.595 

30 

.074 

-074 

40 00 

68.982 

4.422 

8.844 

13.266 

17.688^ 

22.110 

26.532 
















































































COORDINATES FOR PROJECTION OF MAPS, 


42 ; 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale or one inch to one mile. 


Vh 

i u 

Q § £ 

— 2 be . 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 

Latitude c 
Parallel. 

Meridiona 
5^ tances f 
^ Even-de 
Parallel 

5 ' 

long. 

10' 

long. 

15 ' 

long. 

20' 

long. 

25 ' 

long. 

30 ' 

long. 

40°00' 

inches. 

68.982 

inches. 

4.422 

inches. 

8.844 

inches. 

13.266 

inches. 

17.688 

inches. 

22.110 

inches. 

26.532 

IO 

II.499 

4 - 4 TI 

8 822 

13-233 

17.644 

22.055 

26.466 

20 

22.998 

4.400 

8.800 

13.201 

17.601 

22.001 

26.401 

30 

34-497 

4389 

8.779 

13.168 

17-557 

21-947 

26.336 

40 

4^.996 

4 378 

8-757 

‘ 3-135 

I 7 - 5 I 4 

21.892 

26.271 

50 

57-495 

4-368 

8-735 

13-103 

17.470 

21.838 

26.206 

41 OO 

68.994 

4-357 

8.713 

13.070 

17.427 

21.784 

26.140 

IO 

11.501 

4-346 

8.691 

13-037 

I 7-383 

21.728 

26.074 

20 

23 002 

4-335 

8 669 

13.004 

I 7-338 

21.6-73 

26.007 

30 

34-503 

4 324 

8.647 

12.971 

17.294 

21 6l8 

25-941 

40 

46.004 

4 - 3 12 

8.625 

12.937 

i 7 - 25 ‘ 

21.562 

25-875 

50 

57 - 5 o 6 

4.301 

8 603 

12.904 

17.20=; 

21.507 

25.808 

42 OO 

69.007 

4.290 

8.581 

12.871 

17.161 

21.451 

25-742 

IO 

n.503 

4.279 

8 558 

12.837 

17.116 

21-395 

25 674 

20 

23.006 

4.268 

8 535 

12.803 

17.071 

21.338 

25 6c6 

30 

34-510 

4.250 

8.513 

12.769 

17.025 

21.282 

25-538 

40 

46.013 

4-245 

8.490 

12-735 

16.980 

21.225 

25.470 

50 

57-516 

4-234 

8.467 

12.701 

16.935 

21.169 

25.402 

43 00 

69.019 

4.222 

8-445 

12.667 

16.890 

21. 112 

25-334 

10 

ir. 505 

4.211 

8.422 

12.633 

16.844 

21.054 

25.265 

20 

23.010 

4.109 

8-399 

12.598 

16.798 

20.997 

25.196 

30 

34-515 

4.188 

8.376 

12. 564 

16.751 

20.939 

25.127 

40 

46.020 

4.176 

8-353 

12.529 

16.705 

20.882 

25.058 

50 

57-525 

4.165 

8.330 

12.494 

16.659 

20.824 

24.989 

44 00 

69.030 

4 153 

8 307 

12.460 

16.613 

20.767 

24.920 

IO 

11.507 

4.142 

8.283 

12.425 

16.566 

20.708 

24.849 

20 

23.014 

4.130 

8.260 

I2.39O 

16.519 

20.649 24.779 

30 

34-522 

4.118 

8.236 

12 354 

i 6.473 

20.591 

24.709 

40 

46.029 

4.106 

8.213 

12.319 16.426 

20.532 

24.638 

50 

57-536 

4-095 

8.189 

12.284 16.379 

20.473 

24.568 

45 00 

69.043 

4.083 

8.166 

12.249 

16.332 

20.415 

24 498 


ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 

-a rt 

3 > 



— u 

•—« <D 

bl 

0 

0 

1 

4 i° 

c c 
c ►-< 

*4 



inches. 

inches. 

5 ' 

0.002 

0.002 

10 

.008 

.008 

15 

.019 

.019 

20 

•033 

•033 

25 

.052 

.052 

30 

.074 

.075 


42 ° 

43 ° 

5 

0.002 

0.002 

10 

.008 

.008 

15 

.OI9 

.019 

20 

•033 

•033 

25 

.052 

.052 

30 

.075 

•075 


44 ° 

45 ° 

5 

0.002 

0.002 

10 

.008 

.008 

15 

.019 

.019 

20 

•034 

•034 

25 

-052 

-053 

30 

• 075 

.076 














































































428 


MA P CONS 7 R U C TION. 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale 63¥6(J' or one i nc h to one mile. 



t/3 qj 

Qg« 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 




Vh 

„ ? tuO .. 







ORDINATES OF 


re d v 
c ^ v 







DEVELOPED 

'a 2 

.2 v c — 







PARALLEL. 

cd 

■*-» 

cd •— 1 

•r c > x 

5 ' 

10 

15 ' 

20 

25 ' 

30 ' 


dp 



s 

dl 

long. 

long. 

long. 

long. 

long. 

long. 





inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

OJ —• 
tj cd 
z > 



45°oo / 

69.043 

4.083 

8.166 

12.249 

16.332 

20.415 

24.498 

f J •—1 

•— <l) 

b/)£ 

C C 

c 

45 ° 

46 ° 

10 

I 1.509 

4.071 

8.142 

12.213 

16.284 

20.355 

24 426 

■J 



20 

23.018 

4-059 

8.1x8 

12.177 

16.236 

20.295 

24-354 


inches. 

inches. 

30 

34 528 

4.047 

8.094 

12.141 

16.188 

20.236 

24.283 




40 

46.037 

4-035 

8 070 

12.105 

16.141 

20.176 

24 211 

5 ' 

0.002 

0.002 

50 

57-546 

4.023 

8.046 

12.070 

16.093 

20 . Il6 

24.139 

10 

• OOS 

.008 









15 

.O19 

.019 

46 00 

69-055 

4.on 

8.023 

12.034 

16.045 

20.056 

24.068 

20 

•034 

.034 









25 

•053 

•053 

10 

11.511 

3-999 

7.998 

II.997 

i 5 997 

19.996 

23-995 

30 

.076 

.076 

20 

23.023 

3-987 

7-974 

II.961 

15 948 

19-935 

23.922 




30 

34-534 

3-975 

7-950 

II 925 

15 899 

19.974 

23.849 




40 

46.045 

3-963 

7.925 

11.888; 15.851 

19-813 

23.776 




50 

57-557 

3-951 

7.901 

11.852 ic.802 

19 753 

23.703 




47 00 

69.068 

3-938 

7.877 

11.815 15 754 

1 

19.692 

23.630 


47 ° 

4S° 

10 

II- 5 I 3 

3.926 

7.S52 

11.778 15704 19.630 

23-556 

5 

0.002 

0.002 

20 

23.027 

3 914 

7.827 

11.741 

15 655 

19 569123.482 

10 

.00S 

.008 

30 

34-540 

3.901 

7.803 

11.704 15.606 

19.507123.408 

15 

.019 

.019 

40 

46.053 

3.889 

7-778 

11.067 15-556 

19 445 

23-334 

20 

.034 

•033 

50 

57 567 

3 • S 7 7 

7-753 

11.630,15-507 

1 

19 383 23.260 

25 

•052 

•052 

48 00 

69.080 

3.864 

7.729 

n -593 

15-457 

19-322 23.186 

30 

•075 

•075 

10 

11.516 

3-852 

7.704 

11 555 

15 407 

19.259 

23 hi 




20 

23.031 

3-839 

7.679 

11.518 

15-357 

19 196.23 035 




30 

34-546 

3.827 

7.653 

11.480 

I 5-307 

I 9 -I 34 

22 960 




40 

46.062 

3.814 

7.628 

11.442 

15-257 

19.071 

22.885 


— 


50 

57-577 

3.802 

7.603 

n.405 

15 206 

19.008 22.810 

1 


49 ° 

50 ° 

49 00 

69.093 

3-789 

7-578 

11.367 

15.156 18.945 

22 734 




i 

■ ■ > . 







5 

0.002 

0.002 

10 

II-5I7 

3-776 

7-553 

11.329 

15-105 

18.882 

22.658 

10 

.008 

.008 

20 

23-035 

3 764 

7 527 

11.291 

15-054 

18.818 22.5S1 

15 

.019 

.019 

30 

34.552 

3-751 

7.502 

11-253 

15-003 

18.754 

22.505 

20 

•033 

•033 

40 

46.070 

3.738 

7 476 

11.214 

14-952 

18 690 22.429 

25 

•052 

.052 

50 

57-587 

3-725 

7-451 

11.176 14 901 

» 

18.627 

22.352 

30 

•075 

.c >75 

;5Q 00 

69.105 

3-713 

7-425 

1 1 

11-138 14.850 18.563 

22.276 



j 









































































COORDINATES FOR PROJECTION OF MAPS. 


429 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale ^356o> or one i nc h to one mile. 



A U 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 



- i 


Q 5 !: 

~ 2 bp 







ORDINATES 

OF 

0 








DEVELOPED 

3 2 

0 £ 

T 3 l 

5 ' 

10' 

15 ' 

20' 

25 ' 

30' 

PARALLEL. 

•— ctf 

t 5 w£ 



long. 




dp 



s 

long. 

long. 

long. 

long. 

long. 





dl 











inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

0) — * 

^ 5 



5o°oo' 

69.105 

3-713 

7.425 

II.I38 

14-850 

18.563 

22.276 

Z. u. 

c C 

0 — 

50 ° 

51 ° 

10 

11.520 

3.700 

7-399 

I I.O99 

14.799 

1S.499 

22.198 







20 

23.039 

3.687 

7-374 

11.060 

14-747 

lS -734 

22.121 


inches. 

inches. 

30 

34-558 

3-674 

7.348 

II .021 

14 695 

18.369 

22.043 

r' 

0.002 
on 8 

0.002 
no 8 

40 

46.078 

3.661 

7.322 

IO.983 

14 644 

18.305 

21.965 

5 

TO 

50 

57-598 

3.648 

7.296 

10-944 

14-592 

18.240 

21.888 

15 

.019 

.019 

51 00 

69.117 

3-635 

7.270 

IO.905 

14.540 

18.176 

21.811 

20 

25 

-033 

.052 

•033 

.051 

10 

20 

30 

40 

50 

11.521 
23-043 
34 564 
46.086 
57-607 

3.622 

3.609 

3-596 

3.583 

3-570 

7.244 

7.218 

7 -I 9 I 

7-165 

7-139 

10.866 
10.827 
10 787 
10.748 
10.709 

14.488 

14.436 

14.383 

14-330 

14.278 

18.no 
18.045 

17-979 

I 7 . 9 I 3 

17.848 

21.732 

21.653 

21.574 

21.496 

21.417 

30 

-075 

.074 




52 00 

69 128 

3-556 

7 .II 3 

10.669 

14.226 

17.782 

21.338 


52 ° 

53 ° 

10 

11-523 

3-543 

7.086 

10.629 

14.172 

17.716 

21.259 

5 

0.002 

0 002 

20 

23.047 

3-530 

7.060 

10.589 

I 4 -II 9 

17.649 

21.179 

10 

.008 

.008 

30 

34-570 

3 . 5 i 6 

7-033 

10.550 

14.066 

17 583 

21.099 

15 

.018 

.018 

40 

46.094 

3-503 

7.006 

10.510 

14.OI3 

17.516 

21.019 

20 

•033 

.032 

50 

57 - 6 i 7 

3 49 ° 

6.980 

10.470 

13.960 

17-450 

20.939 

25 

.051 

.050 









30 

-073 

•073 

53 00 

69.140 

3-477 

6-953 

10.430 

13.906 

I 7-383 

20.860 




10 

IT-525 

3-463 

6.926 

10.389 

I3-852 

17.316 

20.779 




20 

23.051 

3.450 

6.899 

10.349 

I3-798 

17 248 

20.69S 




30 

34.576 

3-436 

6.872 

10.309 

13-745 

17.181 

20.617 

20.536 




40 

46.102 

3-423 

6.845 

TO. 268 

13.691 

17.114 




50 

67.627 

69. 152 

3-409 

3-396 

6.818 

6.791 

I 0 . 22 S 

10.187 

13.637 

I3.583 

17.046 

16.979 

20.455 

20.374 


54 ° 

55° 

54 00 

5 

0.002 

0.002 







10 

11.527 

3-382 

6.764 

10.146 

I3.528 

16.910 

20.292 

10 

.008 

.008 

20 

23-055 

3-368 

6-737 

10.105 

13-474 

16.842 

20 210 

15 

.018 

.018 

3° 

34.582 

3-355 

6 709 

IO.064 

I3-4I9 

i6.774 

20 128 

20 

.032 

.032 

40 

46. IO9 

3-341 

6.682 

10.023 

13-364 

16.706 

20.047 

25 

.050 

.049 

50 

57.636 

3.327 

6.655 

9.982 

13.310 

16.637 

19.964 

30 

.072 

.071 

55 00 

69.164 

3-314 

6.628 

9.941 

13-255 

16.569 

19.883 





































































430 


MAP CONSTRUCT1OA 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale or one to one m ^ e> 


Vh 

<2 u 

Q B u 

« ? bi . 

ABSCISSAS OF 

DEVELOPED PARALLEL, dttl 

c 

c« 








C 'O V 







•o — 

3 2 

■- v c ~z 
•a « v 2 

5 ' 

10' 

15 ' 

20' 

25' 

3 o' 

.3 

SO, 

'C E <> « 




long. 




s 

long. 

long. 

long. 

long. 

long. 


dl 








inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

55°oo' 

69.164 

3-314 

6.628 

9.941 

13.255 

16.569 

19-883 

IO 

11.529 

3-300 

6.600 

9.900 

13.200 

16.500 

19.800 

20 

23-059 

3.286 

6.572 

9-859 

13-145 

16.431 

19.717 

30 

34.588 

3.272 

6-545 

9.817 

13.089 

16.362 

19.634 

40 

46.H7 

3-258 

6.517 

9 776 

13 034 

16.293 

19-551 

50 

57.646 

3-245 

6.489 

9-734 

12.979 

16.224 

19.468 

56 00 

69.176 

3-231 

6.462 

9-693 

12.924 

16.155 

I 9-385 

10 

II 531 

3.217 

6-434 

9.651 

12 868 

16.085 

19.301 

20 

23.063 

3-203 

6.406 

9.609 

12.812 

16.015 

19.217 

30 

34-594 

3.189 

6.378 

9-567 

12.756 

15-945 

19-134 

40 

46.125 

3-175 

6.350 

9-525 

12.700 

15.875 

19 050 

50 

57-656 

3.161 

6.322 

9-483 

12.644 

15.805 

18.966 

57 00 

69.188 

3-147 

6.294 

9.441 

12.588 

15-735 

18.882 

10 

H -533 

3-133 

6.266 

9-398 

12.531 

15.664 

18 - 797 , 

20 

23.066 

3 .II 9 

6.237 

9 356 

12-475 

15.594 18.712 

30 

34-599 

3.104 

6.209 

9.314 

12.418 

15-523 

18.627 

40 

46.132 

3.090 

6.181 

9.271 

12 362 

15-452 

lS.542 

50 

57-666 

3 076 

6.152 

9.229 

12.305 

15-381 

18.457 

58 00 

69.199 

3.062 

6.124 

9.186 

12.248 

I 5 - 3 II 

18.373 

10 

n -535 

3-048 

6.096 

9 M 3 

12.191 

15-239 

lS.287 

20 

23.074 

3034 

6.067 

9.101 

12.134 

15.168 

18.201 

30 

34-605 

3.019 

6.038 

9 058 

12.077 

15.096 18 115 

40 

46.140 

3-005 

6.010 

9.015 

12 020 

15.025 

18.029 

50 

57-675 

2.991 

5.981 

8.972 

11.962 

14-953 

17-944 

59 00 

69.210 

2 976 

5-953 

8.929 

11.905 

14.882 

17-858 

10 

n -537 

2.962 

5-924 

8.885 

11 847 

14.809 

17-771 

20 

23.074 

2.947 

5 895 

8.842 

11.790 

1 - 4-737 17-684 

30 

34.610 

2-933 

5.866 

8.799 

11.732 

14.665 

17-597 

40 

46.147 

2 918 

5-837 

8.755 

11.674 

14-562 

17.510 

50 

57.684 

2.904 

5.808 

8.712 

11.616 

14.520 

17.424 

60 00 

69.221 

2.890 

5-779 

8.669 

11.558 

14 448 17-337 

1 










ORDINATES OF 
DEVELOPED 
PARALLEL. 

dp 


<v — 
Tj ft 

2 t 

‘b£« 
c c 
o*- 1 

hJ 


5 

IO 

15 

20 

25 

30 


5 

IO 

15 

20 

25 

30 


5 

io 

15 

20 

25 

30 


55 ° 

56 ° 

inches. 

inches. 

0.002 

0.002 

.008 

.008 

.018 

.018 

.032 

.031 

•049 

.049 

.071 

.070 

57 ° 

58 ° 

0.002 

0.002 

.008 

.008 

.017 

.017 

.031 

.030 

.048 

•047 

.069 

.068 

59 ° 

O' 

0 

0 

c 002 

0.002 

.007 

.007 

.017 

.016 

.030 

.029 

.046 

•045 

.067 

.065 






































































COORDINATES FOR PROJECTION OF MAPS, 


43 * 


Table XXIII. 


COORDINATES FOR PROJECTION OF MAPS. 

Scale £3^6$, or one inch to one mile. 



QSi! 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 




V-t 

— < 2 <r> 







ORDINATES 

OF 









DEVELOPED 

• 0 = 

5 2 

•2 S c-3 

5' 

10' 

15' 

20 ' 

25 ' 

30 ' 

PARALLEL. 

a! 

*2- 

7" 5 

V 2WCL. 




dp 

1 


2 

long. 

long. 

long. 

long. 

long. 

long. 





dl 


















u— • 




inches. 

inches. 

inches. 

inches. 

inches. 

inches, j 

inches. 

P > 



6o°oo' 

69 221 

2.890 

5.779 

8.669 

11-558 

14.448 

17-337 

Sue" 

C a 

o H 

6o° 

6i° 

10 

n -539 

2.875 

5.750 

8.625 

II.500 

14-375 

17.249 

HH 






20 

23.077 

2.860 

5-721 

8.581 

H-44I 

14.302 

17.162 


inches. 

inches. 

30 

40 

34.616 

46.154 

2.846 

2.831 

5 - 69 T 

5.662 

8-537 

8-493 

11.383 

11.324 

14.229 

14-156 

17.074 

16.987 

5 ' 

0.002 

0.002 

50 

57-693 

2.8l6 

5-633 

8.450 

11.266 

14.083 

16.899 

IO 

15 

. 007 
.016 

.007 

.016 

61 00 

69.232 

2.802 

5.604 

8.406 

ir. 208 

14.010 

l6.8ll 

20 

25 

.029 

•045 

.029 

•045 

10 

20 

30 

40 

50 

11.540 
23.081 
34.621 
46.162 
57-702 

2.787 

2.772 

2.758 

2-743 

2.728 

5-574 

5-545 

5 -II 5 

5-486 

5-456 

8.361 
8.317 
8-273 
8.229 
8.184 

11.148 
11.090 
11.030 
10.972 
10.912 

I 3-936 
13 863 
13 788 
I 3 - 7 I 5 
13.641 

16.723 

16.634 

16.546 

16.457 

16.369 

30 

.065 

.064 




62 00 

69.242 

2.713 

5.427 

8.T40 

10.854 

I 3-567 

16.280 


62° 

63 ° 

10 

11.542 

2.699 

5-397 

8.096 

10.794 

13 493 

16.191 

5 

0.002 

0.002 

20 

23.084 

2.684 

5-367 

8.051 

10.734 

13-418 

l6.102 

10 

.007 

.007 

30 

34.626 

2.669 

5-337 

8.006 

10.675 

13-344 

[6.012 

15 

.016 

.015 

40 

46.168 

2.654 

5.308 

7.961 

10.615 

13.269 

I 5 . 9 2 3 

20 

.028 

.027 

50 

57 - 7 io 

2.639 

5-278 

7-917 

10.556 

13-195 

j 15 833 

25 

•044 

• 043 







30 

.063 

.061 

63 00 

69-253 

2.624 

5.248 

7.872 

10.496 

13.120 

15-744 




10 

n -544 

2.609 

5.218 

7.827 

10.436 

13-045 

15.654 




20 

! 23.087 

2-594 

5.188 

7.782 

10.376 

12.970 15-564 




30 

34-631 

2-579 

5.158 

7-737 

10.316 

12.895 15-473 




40 

50 

64 00 

‘46.175 

57-718 

69.262 

2.564 

2 549 

5-128 
5.09S 

5.068 

7.692 

7.647 

7.602 

10.256 
10.196 

10.136 

12.820 

12-745 

12.670 

i 5 - 3»3 
15 293 

15.203 


64 ° 

65 ° 

2-534 

5 

0.002 

0.002 






10 

11.545 

2.519 

5-037 

7.556 

10.075 

12.594 

15-112 

10 

.007 

.006 

20 

23-091 

2.504 

5.007 

7 - 5 i 1 

10 014 

12.518 

15.022 

15 

.015 

.014 

3 ° 

34.636 

2.488 

4-977 

7.465 

9-954 

12.452 

14.930 

20 

.026 

.o ?6 1 

40 

46.182 

2-473 

4-947 

7.420 

9-893 

12.367 14.840 

25 

.041 

.040 

50 

57.727 

2 458 

4.916 

7-374 

9.832 

12.291 

14.749 

30 

.060 

.058 

65 00 

169.272 

2-443 

4.886 

7-329 

9.772 

12.215 

14.658 

1 



























































































432 


MA P CONS TR UC 7 'ION. 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale giifio. or one inch to one mile. 


Latitude of 
Parallel. 

Meridional Dis- 
5^ tances from 
^ Even-degree 
Parallels. 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 

ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 

5 ' 

long. 

10' 

long. 

15 ' 

long. 

20' 

long. 

25' 

long. 

3 o' 

long. 


inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

u 

3 cfl 



65°oo' 

69.272 

2-443 

4.886 

7.329 

9.772 

12.215 

14 658 

'Sob 

c 

0 c 

65° 

66° 

IO 

H -547 

2.428 

4-855 

7.283 

9 - 7 II 

12.139 

14 566 

►J 1-1 



20 

23.094 

2.412 

4-825 

7-237 

9-650 

12.062 

14.474 


inches. 

inches. 

30 

34 - 64 I 

2-397 

4 794 

7.19I 

9-588 

II.9S6 

14 383 




40 

46.188 

2.382 

4.764 

7-145 

9-527 

II.909 

14.291 

5 

0.002 

0.002 

50 

57-735 

2.366 

4 733 

7.100 

9.466 

11-833 

14.199 

10 

.006 

.006 









15 

.014 

.014 

66 oo 

69.282 

2.351 

4.702 

7-054 

9-405 

II.756 

14.IO7 

20 

.026 

.025 




• 





25 

.040 

•039 

TO 

11.548 

2.336 

4.672 

7 007 

9-343 

II.679 

I 4 -OI 5 

30 

.058 

.056 

20 

23-097 

2.320 

4.641 

6.961 

9.282 

II.602 

13.922 




30 

34.646 

2.305 

4.610 

6.915 

9.220 

11-525 

13-830 




40 

46.194 

2.290 

4-579 

6.869 

9.158 

II.448 

I 3.738 




50 

57.742 

2.274 

4-548 

6.823 

9 097 

II .371 

I 3.645 




67 00 

69.291 

2.259 

4.518 

6.776 

9.035 

II.294 

13 553 


67 ° 

68° 

10 

ii-55o 

2.243 

4 - 487 

6.730 

8-973 

11.217 

13 460 




20 

23.100 

2.228 

4-455 

6.683 

8.911 

H.T39 

13.366 

5 

0.001 

0.001 

30 

34.650 

2.212 

4.424 

6.637 

8.849 

11.061 

13-273 

10 

.006 

.006 

40 

46 200 

2.I97 

4-393 

6.590 

8.787 

10.984 

13.180 

15 

.014 

.013 

50 

57-750 

2.1SI 

4.362 

6-543 

8.724 

10.906 

13-087 

20 

.024 

.023 









25 

.038 

.036 

68 00 

69.300 

2. 166 

4-331 

6-497 

8.662 

10.828 

12.994 

30 

.054 

.053 

10 

n-552 

2.150 

4.300 

6.450 

8.600 

10.750 

12.900 




20 

23.103 

2-134 

4.269 

6.403 

8.538 

10.672 

12.806 




30 

34-654 

2.II9 

4-237 

6.356 

8-475 

10.594 

12.712 




40 

46.206 

2.IO3 

4.206 

6.309 

8 412 

10.516 

12.619 




50 

57-753 

2.088 

4-175 

6.263 

8.350 

10.438 

12.525 


69 ° 

70 ° 

69 00 

69.309 

2.072 

4.144 

6.216 

8.288 

10.360 

12.431 












5 

0.001 

0.001 

10 

11-553 

2.056 

4.112 

6.169 

8.225 

10.281 

12-337 

10 

.006 

.005 

20 

23.106 

2.O4O 

4.081 

6.121 

8.162 

10.202 

12.242 

15 

.013 

.012 

30 

34-659 

2.025 

4.049 

6.074 

8.099 

10.124 

12.148 

20 

.022 

.022 

40 

46.212 

2.009 

4.018 

6.027 

8.036 

10.045 

12.054 

25 

• 035 

•034 

50 

57.764 

1-993 

3-986 

5 - 98 o 

7 973 

9.966 

n -959 

30 

.051 

.049 

70 00 

69.317 

1-977 

3-955 

5-932 

7 . 9 io 

9.888 

11.865 








































































COORDINATES FOR PROJECTION OF MAPS. 


433 


Table XXIII. 


COORDINATES FOR PROJECTION OF MAPS. 
Scale 63^0’ or one i nc h to one mile. 


Latitude of 

Pa rallel. 

Meridional Dis- 
^ tances from 
Even-degree 
Parallels. 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 

ORDINATES OF 

DEVELOPED 

PARALLEL. 

dp 

5 ' 

long. 

to' 

long. 

15 ' 

long. 

20' 

long. 

25' 

long. 

30 ' 

long. 


inches. 

inches 

inches. 

inches 

inches. 

inches. 

inches. 

<u—' 

X) ™ 

3 £ 



7o°oo‘ 

69.317 

1 - 9 . / 

3-955 

5-932 

7.910 

9.888 

II.865 

C c 

O’ -1 

70 ° 

7 i° 

TO 

II. ^ 

1.96° 

3 • Q23 

5-885 

7 . 846 

9.808 

II.770 




20 

23.109 

1.946 

3.892 

5.837 

7-783 

9.729 

H.675 


inches 

inches. 

30 

34-663 

1.930 

3.860 

5.790 

7.720 

9-650 

H -579 

_ f 



40 

46.217 

1.914 

3.828 

5.742 

7.656 

9-571 

11.485 

5 

0.001 

0.001 

50 

57-772 

1.898 

3-796 

5-695 

7-593 

9.491 

11.389 

10 

.005 

. 005 









15 

.012 

.012 

7i oo 

6q.326 

1.882 

3-765 

5-647 

7-530 

9.412 

11 . 294 

20 

.022 

.021 









25 

•°34 

•032 

IO 

it -556 

1.866 

3-733 

5.600 

7.466 

9-333 

11 -199 

30 

.049 

•047 

20 

23.TII 

1.850 

3.701 

5-552 

7.402 

9-253 

n . 103 




30 

34.667 

1-835 

3.669 

5.504 

7-338 

9-173 

11.008 




40 

46.222 

1 . 8 T9 

3-637 

5-456 

7-275 

9.094 

10.912 




50 

57-778 

1.803 

3.605 

5.408 

7 . 211 

9.014 

10.816 




72 00 

69-334 

H-t 

OO 

^4 

3-574 

5 - 36 o 

7-147 

8-934 

10.721 


72 ° 

73 ° 

10 

11.557 

1.771 

3-542 

5-312 

7.083 

8.854 

10.625 

5 

0.001 

0.001 

20 

23.114 

1-755 

3-509 

5 264 

7.019 

8.774 

10.528 

10 

.C05 

.005 

30 

34.670 

1-739 

3-477 

5.216 

6-955 

8.694 

10.432 

15 

.Oil 

.011 

40 

46.227 

1.723 

3-445 

5.168 

6.89T 

8.614 

10.336 

20 

.020 

.019 

50 

57-784 

1.707 

3-413 

5-120 

6.826 

8-533 

10.240 

25 

•031 

.029 









30 

•044 

.042 

73 00 

69-341 

r. 691 

3-38i 

5.072 

6. 762 

8-453 

10.144 




TO 

11.558 

1.674 

3 • 349 

5.024 

6.698 

8-373 

10.047 




20 

23.116 

1.658 

3-317 

4.975 

6.634 

8.292 

9-950 




30 

34-674 

1.642 

3.284 

4.927 

6.509 

8.211 

9-853 




40 

46.232 

1.626 

3.252 

4.878 

6.504 

O . I j I 

9-757 




50 

57-790 

1.610 

3.220 

4.830 

6.440 

8.050 

9.660 


74 ° 

75 ° 

74 00 

69.348 

1-594 

3-188 

co 

^1- 

6.376 

7.970 

9-563 












5 

0.001 

0.001 

10 

tt -559 

i-578 

3-155 

4-733 

6.311 

7.889 

9.466 

10 

.004 

.004 

20 

23.118 

1.562 

3.123 

4-685 

6.246 

7.808 

9-369 

15 

.010 

.009 

3 ° 

34-677 

1-545 

3.091 

4.636 

6.181 

7.727 

9.272 

20 

.018 

.017 

40 

46.236 

1.529 

3.058 

4-587 

6.116 

7-645 

9 1 75 

25 

.028 

.026 

50 

57.796 

1 .513 

3.026 

4.539 

6.052 

7.565 

9 -o .77 

30 

.040 

038 

0 

0 

in 

69.355 

1.497 

2-993 

4.490 

5-987 

7.484 

8.980 




















































































434 


MAP CONSTRUCTION. 


Table XXIII. 

COORDINATES FOR PROJECTION OF MAPS. 
Scale or one inch to one mile. 



n a « 

ABSCISSAS OF 

DEVELOPED PARALLEL, dm 




V*H 

o • 

_ O b£ • 







ORDINATES OF 

u'v 

as 4 : 

a " 







DEVELOPED 

•0-3 

3 2 

.2 ^ 

T 3 ° a; 2 

5 ' 





30' 

PARA 1 LEL. 

. 3 ! rt 

T “ ^ CTj 

v a. 

10 

15 

20 

25 


dp 


-] 

S 

dl 

long. 

long. 

long. 

long. 

long. 

long. 





inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

v — 

-a 03 

3 > 



75 °oo' 

69-355 

I.497 

2 • 993 

4.490 

5-987 

7.484 

8.980 

*-» u 

uH 

C c 

C 

75 ° 

76 ° 

IO 

t r.560 

1.480 

I.464 

2.961 

2.928 

4.441 

5.922 

5.856 

7.402 

7.321 

8.882 




20 

23.120 

4-392 

8.785 


inches. 

inches 

30 

34.681 

I.448 

2.896 

4-344 

5-792 

7.240 

8.687 

5 ' 



40 

46.241 

I.432 

2.863 

4-295 

5-726 

7-158 

8.590 

0.001 

0.001 

50 

57-8or 

I- 4 I 5 

2.831 

4.246 

5.661 

7.077 

8.492 

10 

.004 

.004 









15 

.009 

.009 

.016! 

76 00 

69.361 

1-399 

2.798 

4.197 

5.596 

6-995 

8-394 

20 

.017 







25 

30 

. 026 

. 02s J 

10 

11-561 

1-383 

2.765 

4.148 

5-530 

6.913 

8.296 

.038 

.036! 

20 

23.122 

1.366 

2-733 

4.099 

5-465 

6.832 

8.198 




30 

34 -6S3 

i- 35 o 

2.700 

4-050 

5-400 

6.750 

8.099 




40 

46.244 

1-334 

2.667 

4.001 

5-334 

6.668 

8.002 




50 

57-806 

I- 3 I 7 

2.634 

3-952 

5.269 

6.586 

7-903 

7-805 




77 00 

69.367 

1.301 

2.602 

3-903 

5-204 

6 505 


77 ° 

78° : 












10 

11.562 

1.284 

2.569 

3-854 

5 -I 38 

6.423 

7.707 

5 

0.001 

0.001 

20 

23.124 

1.268 

2.536 

3.804 

5.072 

6.341 

7.609 

10 

.004 

• 003 

30 

34.686 

I . 252 

2.503 

3-755 

5.006 

6.258 

7 - 5 io 

15 

.008 

. 008 

40 

46.248 

1-235 

2.470 

3.706 

4.941 

6.176 

7.411 

20 

.015 

. 014 

50 

57 -Sio 

1.219 

2-438 

3-656 

4-875 

6.094 

7-313 

25 

.023 

.021 

78 00 

69-373 

1.202 

2.405 

3.607 

4.810 

6.012 

7.214 

30 

•033 

.031 

10 

20 

30 

40 

11.563 
23.126 
34.689 
46.252 

1.186 
1.169 
I-I 53 
1.136 

2.372 

2-339 
2.306 
2.273 

3-558 
3 - 5 °S 
3-459 
3.410 

4-744 

4.678 

4.612 

4.546 

5-930 

5.847 

5-765 

5-683 

7 -11 5 
7.016 
6.918 
6 819 







50 

79 00 

57-814 

69377 

1.120 

1.104 

2.240 

2.207 

3 - 36 o 

3 - 3 11 

4.480 

4.414 

5.600 

5-518 

6.720 

6.621 


79 ° 

8o° 


0. COI 

0.001 









5 

10 

11.564 

1.087 

2.174 

3.261 

4-348 

5-435 

6. 522 

10 

.003 

• 0031 

20 

23.127 

1.070 

2.141 

3.211 

4. 282 

5-352 

6.422 

15 

.G07 

.106 

30 

34.691 

1.054 

2.108 

3.162 

4.216 

5.270 

6.323 

20 

.013 

• Oil! 

40 

46.255 

1-037 

2.075 

3. rr2 

4.150 

5 -187 

6.224 

25 

.020 

.018 

50 

57.818 

1.021 

2.042 

3.062 

4.083 

5.104 

6.125 

30 

.028 

.026 

80 00 

69.382 

1.004 

2.009 

3 -°i 3 

4.017 

5.022 

6.026 




mm -- 










































































































USE OF PROJECTION TABLES . 


435 


186. Use of Projection Tables —Where it is proposed to 
project a map on a scale which bears a decimal ratio in inches 
to linear miles, the quantities to be laid off can be derived 
directly from Table XXIII. This table is arranged on the 
scale of one mile to one inch, and the quantities to be laid 
off for meridians or parallels are given in inches. For any 
other scale, as that of two miles to one inch, and, for ex¬ 
ample, for a 30' projection between latitudes 31 0 and 31 0 30', 
and say in longitude 98° to 98° 30' (Fig. 134), the quantities 
to be laid off on the projection are to be obtained in inches 
from the table for every 5' by halving the amounts in the 
table. Quantities required for projections ruled at shorter 
intervals than 5' may be obtained by moving the decimal 
point. Thus for parallels 3' apart the quantity corresponding 
to differences in latitude of 30' is sought and the decimal 
point moved one place to the left, etc. 

Where it is desired to make a projection on any other 
scale than that bearing an even decimal relation of inches to 
miles, projection tables, XXIV, XXV, and XXVI, should 
be used. The first of these, Table XXIV, gives the exact 
lengths of degrees of parallels and meridians in meters and 
in statute miles, and these may be reduced to inches or 
other scale. Tables XXV and XXVI may be used in pro¬ 
jecting large-scale maps, approximately within the limits of 
the United States, between latitudes 24 0 and 51 0 north. 
The first of each pair of columns in Table XXV gives the 
latitude, and opposite to it the corresponding length of one 
minute of parallel in meters. These may be reduced to any 
map scale by consultation of reduction tables (Chap. XXX.) 
Corresponding values less than one minute may be obtained 
by moving the decimal point one place, which will give the 
value for six seconds. Thus, in Table XXV, for latitude 28° 
the length of one minute of parallel is 1639.4 meters. The 
length of six seconds of the same parallel is obtained by mov¬ 
ing the decimal point one place to the left, 163.94 meters. 


MAP CONSTRUCTION. 


436 

For the lengths of meridional arcs the quantities dm are 
obtained for a given latitude from Table XXVI in the follow- 
ing manner: For the latitude and for the number of degrees 
of longitude included in the projection, the length of dm as 
given in meters, which is to be found in the first column, 
is to be laid off both to right and left of the vertical central 
meridian. At each of the points thus found perpendiculars 
are to be erected which will be parallel to the central meridian, 
and the lengths of the corresponding ordinates dp are to be 
laid off upon them. Through the extremities of each of these 
perpendiculars draw lines which will give the confining out¬ 
lines of the curves of the parallels and meridians. Spaces 
between the extremities dp may now be divided into con¬ 
venient equal parts of the same value, 5' or 15', etc., as was 
given the spaces between the meridians. Curved lines drawn 
between these will represent the parallels of the completed 
projection according to the number of equal parts used. 

187. Areas of Quadrilaterals of Earth’s Surface. —It is 
sometimes desirable to determine the areas of quadrilaterals 
of the earth’s surface, and these may be found directly from 
Table XXVII. Areas of quadrilaterals of less or greater ex¬ 
tent than one degree may be found by simple division or 
multiplication. 

188. Platting Triangulation Stations on Projection.— 

The projection of the map being now constructed, it is neces¬ 
sary to plat upon it the exact positions of the triangulation 
stations. These must, of course, have been previously com¬ 
puted, so that their geodetic coordinates (Chap. XXIX) are 
exactly known. These coordinates are given in degrees, 
minutes, and seconds of arc. Assume that the projection has 
been so platted that meridian and parallel lines are shown for 
every ten minutes; then the nearest degrees and ten minutes 
of latitude and longitude of each position are taken out and 
the corresponding rectangle found in which the point will fall. 
The odd minutes and seconds, those greater than ten min- 


PROJECTION TABLES. 


437 


in 

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CM 

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P 

JP 


I s These quantities express the number of meters and statute miles contained within an arc of which the decree of latitude named is the middle ; 
i, the quantity, 111,032.7, opposite latitude 40°, is the number of meters between latitude 39 0 30' and latitude 40° 30'. 




















































































43« 


MAP CONSTRUCTION . 


I 


Table XXV. 

FOR PROJECTION OF MAPS OF LARGE AREAS. 

(Extracted from Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1884.) 
ARCS OF THE PARALLEL IN METERS. 


Latitude. 

Valueof 1'. 

Latitude. 

Valueof i'. 

Latitude. 

Valueof i'. 

Latitude. 

i 

Valueof 1'. 

24° 00' 

1695.9 

3 t° OO' 

I 59 I -8 

38° OO' 

1463.9 

45 00 

1314.2 

IO 

3-7 

IO 

89.0 

IO 

60.6 

10 

IO.3 

20 

1-5 

20 

6.2 

20 

57-3 

20 

06.5 

30 

1689.3 

30 

3-4 

30 

53-9 

30 

02.7 

40 

7.0 

40 

0.6 

40 

50.6 

40 

1298•8 

50 

4.8 

50 

77-8 

50 

47.2 

50 

95-0 

25 OO 

1682.5 

32 OO 

1574-9 

39 00 

1443.8 

46 00 

1291.O 

10 

80.3 

IO 

72.1 

10 

40.4 

10 

87.2 

20 

1678.0 

20 

69. 2 

20 

37 -o 

20 

83-3 

30 

5-7 

30 

6-3 

30 

33-6 

30 

79-4 

40 

3-3 

40 

3-4 

40 

30.2 

40 

75-5 

50 

1.0 

50 

o -5 

50 

26.7 

50 

71.6 

26 OO 

1668.7 

33 00 

1557-6 

40 00 

I 423-3 

47 00 

1267.6 

IO 

6-3 

10 

4-7 

10 

19.8 

10 

63-7 

20 

3-9 | 

20 

i -7 

20 

16.3 

20 

59-7 

30 

i -5 

30 

48.7 

30 

12.8 

30 

55-8 

40 

1659.1 ! 

40 

5-8 

40 

09-3 

40 

51.8 

50 

6.7 

50 

2.8 

50 

05.8 

50 

47.8 

27 OO 

1654-3 

34 00 

1539-8 

41 00 

1402.3 

4s 00 

1243.8 

IO 

51-8 

10 

6.8 

10 

1398.8 

10 

39-8 

20 

1649.4 

20 

3-7 

20 

95-2 

20 

35-8 

30 

6.9 

30 

0.7 

30 

91.6 

30 

3 i -7 

40 

4.4 

40 

27.6 

40 

88.1 

40 

27.7 

50 

1.9 

50 

4.6 

50 

84-5 

50 

23.6 

28 OO 

1639.4 

35 00 

1521.5 

42 00 

1380.9 

49 00 

t 219.6 

IO 

6.9 

10 

18.4 

10 

77-3 

10 

15-5 

20 

4-3 

20 

15-3 

20 

73-7 

20 

11.4 

30 

1.8 

30 

12.2 

30 

70.0 

30 

07-3 

40 

29.2 

40 

09.1 

40 

66.4 

40 

03.2 

50 

6.6 

50 

° 5-9 

50 

92.7 

50 

H99.1 

29 OO 

1624.0 

36 00 

1502.8 

43 00 

I 359 -I 

50 00 

ii 95 -o 

TO 

21.4 

10 

1499.6 

10 

55-4 

10 

90.8 

20 

18.8 

20 

6.4 

20 

5 i -7 ! 

20 

86.7 

30 

6.1 

30 

3-2 

30 

48.0 | 

30 

82.5 

40 

3-5 

40 

0.0 

40 

44-3 

40 

78.4 

50 

08 

50 

86.8 

50 

40.5 

50 

74-2 

30 OO 

1608.1 

37 00 

1483.6 

44 00 

1336.8 



IO 

5-4 

10 

80.3 1 

10 

33 -1 



20 

2.7 

20 

77 -i 

20 

29-3 



30 

0.0 

30 

73-8 

30 

25-5 



40 

1597-3 

40 

70.5 

40 

21.7 



50 

4-5 

50 

67.2 

50 

18.0 
































































Table XXVI. —for projections of maps of large areas. 

(Extracted from Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1884, by Henry Gannett.) 
MERIDIONAL ARCS. COORDINATES OF CURVATURE. 

NATURAL SCALE—VALUES OF dm AND dp IN METERS. 


PROJECTION TABLES 


439 




ro 

(N 

r>. 

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00 m m m 

mow O ■t 

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04 N O' O CO 

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00 

cn 







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Table XXVI. —for projections of maps of large areas. 

(Extracted from Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1884, by Henry Gannett.) 

MERIDIONAL ARCS. COORDINATES OF CURVATURE. 


PROJECTION TABLES. 


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m G ~t- t^oo 

00 0 

CC NO 

0 

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moo O' 

00 vo vo m Ns 


3 

ft 

-fOO 

M VO 

0 "Too <m m 

O envo O' cm" 

CM* 00 " 

m envo 

00 

O' 

Hi 

cm m 

- 1 - m 



*-» 

V- 

00 VO 

m co 

cm 0 oo m 

CC pi 0 CO tv 

m m 

cm 0 co 

0 

-r 

m 

- 0 

Ns in 


5d 

' ~ 



*-• 

N CO 

Tt- m 

mvo 

00 O' 0 0 ^ 

cm cn 

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m 

0 

t^oo 

O O' 

O w 

cm m m m#- 

X 

03 







>- M M 

►“« W 

M M 

HI 


»H 

M 

M P4 

CM CM' 

cm cm" cm cm 


x 

















>—* 


















< 




0 

0 0 

0 0 

0 Q 0 

0 0 0 0 0 

0 0 

0 0 

0 

0 

0 

Q 

0 0 

0 O 

0000 

> 



8 

0 

0 0 

0 0 

O 0 0 

OOOOO 

0 0 

0 0 

0 

0 

0 

0 

0 O 

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0000 

1 


o 

o 















1 


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N 

CO 

mvo 

r^oo O' 

O w CM CO 't 

mo 

t^oo 

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0 

w 

CM 

m f 

mvo 

Ns OO O' 0 

w 









>-> M M 

HI 

CM 

CN 

CM 

CM CM 

CM CM 

CM CM CM cn 

1—1 

< 



O'VO 

M CO 

CM 00 

0 t^OO 

cm *0 m m m 

>o 0 

Nn m 

CM 

m 

CM 

h* 00 CM 

G 00 


f) 



Ns 

M 

M VO 

l''- COVO COvO 

moo c- m O' 

CN O 

m 

C^. 

0 


Ns 

0 m 

m cm 






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m rn 0 O 

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00 

CM O 

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m -if m 0 





















v 



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M C-X 

co o 00 

t^oo 0 cn 

cm 

DO -f- 

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0 

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0 

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M •“> 

CM CO CO 

t mo 00 0 

O (S 

cn m 


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m 

m Ns 

O' CM 

-f Ns O' CM 

H- 









W M 

M M 

HI 

HI 

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CM 

CM CM 

CM CO CO CO CO ^f 

< 

0* 

















cX 

''f 


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0 c 

NOCO N O 

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^ CM 

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00 

moo 

Ns m 

moo 


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m o 

m mvo 

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CN 


O '00 cn 


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T 3 


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— Tf 

O' 

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*- O' VO CM vo 

O Q 

O' 

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O 

N- VO 

CO DO 

0 . 0 in a 0 <s 



•V 

m 

0 vo m 

vo' - 

rC cm rC 

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4 c> 

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mo 

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Ns 00 


< 



00 


in 

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m cn cm 0 0* 

m 

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HI 

O' 


m 

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0 00 


2 

■*-» 




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rt- m 

mvo 

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cm cn 

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0 

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0 0 

Hi H 

cm cn 4 f m 


03 







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H* 

M 

M 

M 

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CM CM 

CM CM CM CM 





















b£ 

c 

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0 

0 

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88 

0 Q 0 

0 0 o 

0 0 0 0 0 

0 0 0 0 0 

8 8 8 8 8 

0 

0 

8 

0 

0 

8 8 

8 8 

8 8 8 8 



o 

o 

M 

C4 

co tJ- 

mvo 

r^oo o 

O h a m + 

mo 

N'OO 

O' 

0 

M 

CM 

CO -N 

mvo 

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H-J 







M M 

HI HI 

” 

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CM 

CM 

CM CM 

CM CM 

CM CM CM cn 




vO 

rr> 

M M 

M M 

0 00 ^ 

t^vo O '0 't- 

CM O 

m cm 

-1- 

r^oo 

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Ns 0 

HI O 





Ns 

0 oo w 

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0 CM O 

CM O' 0 VO vo 

0 

O' -1- 

CN 

m 

N 


co m 

00 CM 




• 

tJ- 

O' 

IN VO 

OO M 

co ^ m 

m -t- cm 0 

CO i- OVO 

HI 

moo 


m -if 

Tf ~f 




§ 


Hh 

■^f t^. 

m rC 

rn 6 oo 

t-^co 0 ro 

vj 4 

cn 

4 

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m 0 

»• r. n » 







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CM CO CO 

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O CM 

m m 

Nh 

00 

0 

CM 

m Ns 

O M 

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o* 









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H 


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CM 

CM CM 

cm m 

cn m m -f 


m 


n> 

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mvo 

cooo oo 

moo vo o' 

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cn 

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m 

Ns 

m hi 





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m cn 

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m O' 

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m 0 

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D 


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CO O' VO 

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m cm oo 

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mod 

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m cm 






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c< co 

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cm m. 

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c>.00 00 

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rn m -f m 









M M « 

M «- 


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H< 

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cm" cm 

CM CM CM* CM 



u 

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0 

c 

0 

0 

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8 8 8 8 8 

0 0 0 0 0 

0 0 0 0 0 

8 8 

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M M 

M 

CM 

CM 

CM 

CM CM 

CM CM 

CM CM CM m 

■ 



























































PROJECTION TABLES , 


443 


if) 

< 

w 

< 

u 

o 

< 


V 

c 

c 

rt 

o 

>* 

u 

c 

<u 

X 

>> 

JQ 


oo 

oo 


fcl* 

o 

in 

cu 

< 

S 

U- 

O 

O) 

z; 

o 

H 

U 

W 

►— 

O 

OS 

cu 

o 


I—1 
> 
X 
X 


t- 

o 

'♦H 


O 

a 

v 

&, 

>» 

4 J 

> 

u 

3 

(/) 


w 

•a 

o 

<u 

O 

■a 

c 

rt 

4-1 

in 

rt 

O 

U 

ISi 

6 

'O 

6 

z 

* 

•a 

c 

V 

CL 

a 

< 

a 

o 

u 


w 
•J 
cc 
< 

H £ 


•o 

HI 

*-» 

y 

Cv 


W 

aS 

D 

H 

C 

> 

aS 

D 

u 

a. 

o 

t/3 

W 

H 

< 

Z 

I—I 

Q 

aS 

O 

O 

u 


t /3 

u 

as 

< 


< 

Z 

o 

HH 

o 

aS 

*-* 





•^■VC m Ci 
oo m m 
t O ro N 

m ^ oo vo 

O' m ov m 

O ^-vo O' - 

O' M Ci 00 00 

Ci O m ov Ov 
m ^ m m m 

^ m ci vo ci 
m m o o> m 
mi- moco 

vo O 00 00 
o x m ci ci 
m x vo x m 

ci tN m m m tN 
tN ci oo t o 
oo 0 Ci Ci ci x 


O 

C 30 

x ^t* tN 

0 » N rn d O' 
h h n m cn 

co oo d - 4 
■^- mvo oo ov 

oo m d "0 m 

0 ci m m 

M M M H 

Ci Ci Ci 4 *vd 

O' x m m cn. 

X Ci Ci Ci Ci 

Ov O' m d o 
o> w t tN o m 

Ci m m m ^ 


Nf 

V 

T 3 


NO O' N N 

o mcf n 
vo ci oo m 

n t nh m 
m m cs 

oo rnvo O' *- 

vo m - o vo 

00 Ci Ci N m 

m ►- ov m o 

N H vo O '00 
vO O' h 0 ) m 

m 't h vo 

X vo 0 X O' 

N N Ci O' N 

00 00 vo O Ci 

O N m m x ci 
tN m x rf mvo 
x tN o O' m cn 


3 

*-* 

+-> 

■§ 

4 - CTn moo 
tN ^ Ci O' 
x Ci Ci 

N n h in o 
h» 't N O' N 
m •t in mvo 

4 -oo 4 - dv 

Tf - O' VO m 
r^oo oo O' 0 

ci iAoo m m 
p* oo m m o 
h - ch rot 

in nov h ci 
^ X O' VO 

Nt mvo vo 

m m 4 - m m d 

mO tN Tf H oo 

oo ov O' o x m 


J 




•H 

M M *-• M M 

M H X IX IX 

X X M ci Ci Ci 



hi 

c 

'8 8 8 8 

Q 0 0 0 Q 

0 0 O 0 0 

0 0 0 0 0 

0 0 0 0 0 

8 8 8 8 8 

8 8 8 8 8 

0 0 0 0 0 0 

0 0 0 0 0 0 



o 

X 

o 

m ci m **• 

mvo t^-oo o* 

O^cim^ 

•H M M M M 

mvo i^oo O' 

M M «1 M M 

Oxcim^ 

Ci N Ci Ci Ci 

mvo NX O' o 

M Ci Ci Ci Ci rn 





• 

in noo >o 
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^* o> m cn 

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mvo ^ 

m ^ 0 N 

NO U )»0 

N- M-iO ^ O' 
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N t ^*00 is M 

oo m moo oo 
oooo smm 

oo m m o m 

x O' 0 m ci 

H N t O' t 

■'t'O vo O' Ci Ov 

Ci Tf 00 rr. 0 vo 

oo x m mvo m 


• 

§ 

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N n 4 h O 
m m ci m m 

oo oo dv ~ 4 * 
^ mvo oo O' 

oo m d"d 4 * 

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m Ci rn t n 

ov x m m tN 

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d m o vo m x 

0 ci m tN o co 
m m m m -r 


o 

















Dd 

w 

V 

T 3 

> 

tN 0 O' M 

m 0 m o 

0 X M X 

■^•^•0 0 0 
m o o h n 

0 O ^ 0 

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Ci O' m ci m 
moo m w m 

tN H OOVO -t- 

^ O'vo moo 
O' M -00 0 Ov 

r^vo n O cv 

OvvO ci oo 

vo x m m 

x m m m cn ci 

m x so oo vo O' 

m ^ x m 

H 

W 

3 

•*-* 

4 

vo « oo r 4 - 
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m M ro 

o' in h n rn 
oo m m o oo 
m '<*■ mvo vo 

go* rn Ov 4 O' 
m m 0 oo m 
r^oo O' O' 0 

moo c 4 ^o r 
m o oo m m 

H N ci CO t 

4 -od x 4 -vo" 

0 n m n o> 
m mvo 

Ov h rn t m>o 

VO Nt X 00 m Ci 

oo O' o o x ci 

s 

CTJ 




M 





) 








z 

HH 


bi> 

8 8 8 8 

8 8 8 8 8 

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8 8 8 8 8 

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0 O 0 O 0 

888888 

$ 


o 

hJ 

o 

x ci rn *• 

mvo t^oo o> 

0 H N fO t 

M M M M « 

mvo t^oo O' 

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0 >-Nmt 

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mvo tN oo ov O 
ci w ci w ci m 










Q 

Z 



vo »on o\ 

00 ^ ^ 

tOvfON 

moo mvo 

m ov h O'-t 
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m m o 

vo ^ O' Q vo 
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m m m t^oo 

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0 O O oo 

■t N O' Ci N 
O' x VO vO 00 
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■*■00 x m ^ m 

§ 

0 * 

NO 

•'t 

nj 

S’ 

W ^ tN 

n m m O' 

m »-• m m m 

oo oo d* c 4 m 
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M M M IX XI 

m m m m tN. 
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x m x in 4 c 4 

0 Ci m tN o m 
m m m m ^ 'vt 

tL, 

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\0 x •'t- m 
■t O' rnN 

^ M ^00 

mvo oo o 

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mvo moo m 

Ci - t^OO T#- 

vo ^ 0 m O' 

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•- xi O' m O' 

m N N N o 
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m m 0 oo m 

r- Ci rc m t 

m ci x 

m o oo m m 
mvo vo c-noo 

4 oo x 4 in O' 

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O' O' 0 x X Ci 

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M M M M IX 

M M X M M 

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8 8 8 8 
o 

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mvo f'soo O' 

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mvo r^oo O' 

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0 x N m rt* 

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25 OO 

26 OO 

27 00 

28 00 

29 00 

30 00 


l-H 









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U 


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O'oo m m m 
m r- o 0 *-■ 

ci m 0"0 ci 
vo m O' O'^t- 
M M VO O'oo 

m x oo m m 

m tN -'*-vo x 
vo m 0 vo Ci 

0 tN tN tN X 00 

O' O' Ci tN ^ X 

VO O tvOOD O' 

CO 

J 


$ 

M 

N tsrOH O' 

m m ci m m 

oo oo O' 4 m 
^ mvo oo O' 

dv 4 * o vo 4 * 

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M M M M M 

m m 4 - moo 

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X VO x In 4 - Ci 

0 d m tN 0 m 
m m m m "T 

< 

o 








OcJ 

X 

•'t 

CU 


n m ov 
Tt-oo oo 

00 vo N 

vo moo Ov 

O'VO mvo oo 

O'VO Ci t -4 -1 

CiO'mNTf 

O' m tj- o 

tCvO t^vO 

^ 0 m 

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O '00 CnVO M 

x m ^ cn m 

0 m t noo 

mO'^O “ Cl 

0 ' 4 - 0 Ci 0 

n 0 “ >1 03 

H 

<3 

3 

iJ 

4 

oo iv.vo in 

Mn m h 
h ci m 

m ci w dsoo 

O' n m m o 
m - 1 * mvo 

vo 4 * ci o oo 
oo vo Ci O' 

t^oo O' o o 

m m o 

Nrnmoso 

m ci m t»- ^ 

h n m c> 4 

vo m x oo vo 
mvo t" i^oo 

d m d 4 *oo x 

Nf x 00 vo m — 

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Z 

S 

j 




m « 

XI M XI x IX 

M X M M X 

X Ci Ci Ci Ci Ci 



bi 

c 

'8 8 8 8 

8 8 8 8 8 

0 0 O 0 0 

0 0 0 0 0 

0 0 O 0 0 

0 0 0 0 o 

8 8 8 8 8 

000000 

000000 



o 

1-4 

o 

m w m ^ 

unvO r^oo o 

o m ci m ^ 

mvo r^oo O' 

M M ix IX IX 

OHdcnt 

Ci Ci Ci Ci Ci 

mvo cnoo o> 0 
ci Ci ci ci Ci m 




vo m moo 
oo t- 
1 - O' co N 

fj vo w m 

m o- m ov 
m -f oo o m 

mvo m ci N 

VC 't O' 0 N 
m r^oo 0 O 

0 ^ m m m 
0 co ci h m 
m o o o 

0 0 Ci Ci vo 
t n t m O' 
m ci O' m o 

ci cn vo m ci x 
tN tN 0 m Ci 0 
m O' m m noo 


o 

§ 

H T#- 

ci t>. m ** <y 
m m ci m m 

co oo O' 4 m 
rf- mvo co O' 

dv 4 - o vo 4 * 

O Ci t m N 

IX IX IX M M 

m m m moo 

O' w m m tN 

X Ci Ci Ci Ci 

X id X IN 4 ^ Ci 

O Ci m in 0 m 

m rn m m tj- rt 


<u 

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3 

4 -< 

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N. 

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lO "* N oo 

0 0 r- 0 

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00 vO rf* W 

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r^.oo t^vo 

6 6 6 6 6 

0 00 vO Tt- Ci 
tj- mvo 

Ci - 0 O' m 

Ci oo - o m 
— vo m mm 

0 dv d'oo 

0 m m -• 

oo oo O' 0 M 

n m h o O' 

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m m O' m m 

vo* m m d 0 * 

O' m m m 

«x ci m m 

vo 0 no 0 m 
O' Tt moo m 
Tt M tN O' O' 

oo~ vo'm cT tN 
00 vO ^ Ci O' 

mvp tNoo oo 

0 m O' 0 tN 0 
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vo 0 _ x ov t m 

4 - X IN Ci 00 rn 
tN m ci 0 in m 

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c 3 

j 




4 M 

M M M M M 

M x M X X 

x Ci <N Ci ci Ci 



b/c 

'o 0 0 0 

I 0 0 0 0 

0 0 0 0 0 

O 0 0 0 0 

0 0 0 0 0 

0 0 0 0 0 

0 0 0 0 0 
0 0 0 0 0 

8 8 8 8 8 

0 0 0 0 0 G 

0 0 0 0 0 0 



o 

1 *-5 

1 o 

H N fn -t 

mo cvco O' 

Oncim-t 

M M M M M 

mvo t^co O' 

M M XI »X x 

Oxtim^ 
Ci Ci Ci Ci Ci 

mvo inoo O' O 

Ci Ci Ci Ci Ci m 


































































444 


MAP CONSTRUCTION. 


Table XXVI. 


FOR PROJECTIONS OF MAPS OF LARGE AREAS. 

(Extracted from Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1884, by 

Henry Gannett.) 

MERIDIONAL ARCS. COORDINATES OF CURVATURE. 



NATURAL SCALE.—VALUES 

OF dm AND dp IN METERS. 

Latitude 49 0 . 

Latitude 50°. 

Long. 

dm . 

dp . 

Long. 

dm . 

dp . 

i° 

OO ' 

73» x 7 2 

• 482 

i° 

oo' 

71,696 

479 

2 

OO 

146,331 

1,928 

2 

OO 

143,379 

1,90 

3 

OO 

219,465 

4,337 

3 

OO 

215,037 

4,3i3 

4 

00 

292,561 

7,709 ' 

4 

OO 

286,656 

7,667 

5 

OO 

365,606 

12,044 

5 

00 

358,224 

1 h ,978 

6 

OO 

438,588 

17,340 

6 

OO 

429,727 

17,246 

7 

OO 

5n,493 

2.3,598 

7 

OO 

5oi,i54 

23,469 

8 

OO 

584,310 

30,815 

8 

00 

572,492 

30,646 

9 

OO 

657,626 

38,991 

9 

OO 

643,727 

38,777 

10 

OO 

729,627 

48,123 

10 

OO 

714,847 

47,859 

II 

OO 

802,102 

58,213 

II 

OO 

785,839 

57,891 

12 

OO 

874,438 

69,254 

12 

CO 

856,691 

68,872 

13 

OO 

946,622 

81,248 

13 

00 

927,389 

80,798 

14 

OO 

1,018,642 

94,191 

14 

00 

997,922 

93,669 

15 

OO 

1,090,485 

108,082 

15 

00 

1,068,277 

M 

O 

00 

10 

l6 

00 

1,162,138 

122,918 

l6 

00 

1,138,440 

122,234 

17 

OO 

i,233,59i 

138,697 

17 

00 

1.208,400 

137,923 

18 

OO 

1,304,829 

I55,4i6 

l8 

00 

1,278,144 

154,546 

19 

OO 

1,375,840 

173,071 

19 

00 

1,347,660 

172,099 

20 

OO 

1,446,613 

191,660 

20 

00 

i,4i6,934 

190,581 

21 

OO 

i,5i7,i35 

211,180 

21 

00 

1,485,956 

209.987 

22 

OO 

1,587,394 

231,623 

22 

00 

i,554,7H 

230,314 

23 

OO 

1,657,378 

252,998 

23 

00 

1,623,189 

251,559 

24 

OO 

1,727,073 

275,288 

24 

00 

1,691,377 

273,717 

25 

OO 

1,796,470 

298,495 

25 

00 

1,759,262 

296,785 

26 

OO 

1,865,554 

322,614 

26 

00 

1,826,833 

320,758 

27 

OO 

i-934,3i5 

347,640 

27 

00 

1,894,077 

345,633 

20 

OO 

2,002,740 

373,570 

28 

00 

1,960,983 

371,404 

29 

OO 

2,070,817 

400,399 

29 

00 

2,027,538 

398,068 

30 

OO 

2,138,536 

428,123 

30 

00 

2,093,731 

425,619 






















PLATTING POSITIONS ON PROJECTION. 445 

Table XXVII. 

AREA OF QUADRILATERALS OF EARTH’S SURFACE OF i° 
EXTENT IN LATITUDE AND LONGITUDE. 

(Prepared by R. S. Woodward.) 


Middle 
Latitude 
of Quad¬ 
rilateral. 

Area 

in Square 
Miles. 

Middle 
Latitude 
of Quad¬ 
rilateral. 

Area 
in Square 
Miles. 

Middle 
Latitude 
of Quad¬ 
rilateral. 

Area 

in Square 
Miles. 

Middle 
Latitude 
of Quad¬ 
rilateral. 

-1 

Area 

in Square 
Miles. 

o° 

OO ' 

4752*33 

25 ° 

30' 

4300.17 

50° 

30' 

3047.37 

75 ° 

3 °' 

1205.13 

o 

30 

52.16 

26 

OO 

4282.50 

51 

OO 

‘ 5*34 

76 

OO 

1x64.49 

I 

OO 

51*63 

26 

3 ° 

64.51 

5 1 

30 

2983.08 

76 

30 

23*75 

I 

30 

50*75 

27 

OO 

46.20 

52 

OO 

50.58 

77 

OO 

1082.91 

2 

OO 

49*52 

27 

3 ° 

27.56 

52 

30 

17*85 

77 

3 ° 

41.99 

2 

30 

47*93 










3 

00 

46.00 

28 

OO 

08.61 

53 

OO 

2884.88 

78 

OO 

1000.99 

3 

30 

43*71 

28 

30 

4189.33 

53 

30 

51.68 

78 

30 

959.90 

4 

OO 

41.07 

29 

OO 

69.74 

54 

OO 

18.27 

79 

OO 

18.73 

4 

30 

38 08 

29 

30 

49 83 

54 

30 

2784.62 

79 

30 

877.49 

5 

OO 

34*74 

30 

OO 

» O 

29.60 

55 

OO 

50.76 

80 

OO 

36.18 

5 

30 

31.04 

30 

30 

4109.06 

55 

30 

16.67 

80 

30 

794*79 

6 

OO 

27.00 

3 i 

OO 

4088.21 

56 

OO 

2682.37 

81 

OO 

53*34 

6 

3 ° 

22.61 

31 

30 

67.05 

56 

30 

47*85 

81 

30 

11.83 

7 

00 

17.86 

32 

OO 

45*57 

57 

OO 

> 3 * l 3 

82 

OO 

670.27 

7 

30 

12.66 

32 

30 

23*79 

57 

30 

2578.19 

82 

30 

28.64 

8 

OO 

07.32 

33 

OO 

ox .69 

58 

00 

43*05 

83 

OO 

586.97 

8 

30 

01.52 

33 

30. 

3979*30 

58 

30 

07 . 70 

83 

30 

45*24 

9 

OO 

4695*38 

34 

OO 

56.59 

59 

OO 

2572.16 

84 

OO 

03*47 

9 

3 ° 

88.89 

34 

30 

33*49 

59 

30 

36.42 

84 

3 ° 

461.66 

io 

OO 

82.05 

35 

OO 

10.28 

60 

OO 

00.48 

85 

OO 

19.81 

TO 

3 ° 

74.86 

34 

30 

3889.67 

60 

30 

2364*34 

85 

30 

377*93 

I I 

OO 

67.32 

36 

OO 

62.76 

61 

OO 

28.02 

86 

00 

36.02 

11 

30 

59*43 

36 

30 

38.56 

61 

30 

2291.51 

86 

30 

294.08 

12 

OO 

51.20 

37 

OO 

14.06 

62 

OO 

54.82 

87 

OO 

52.11 

12 

3 ° 

42.63 

37 

30 

3789.26 

62 

30 

17*94 

87 

30 

10.12 

13 

OO 

33*71 

38 

OO 

64.18 

63 

OO 

2180.89 

88 

OO 

168.12 

13 

3 ° 

24.44 

38 

30 

38.80 

63 

30 

43.66 

88 

30 

126.10 

14 

00 

14.82 

39 

OO 

13. *4 

64 

OO 

06.26 

89 

OO 

84.07 

14 

30 

04.87 

39 

3 ° 

3687.18 

64 

30 

2068.6& 

89 

30 

42.04 

15 

OO 

4594*57 

40 

OO 

60.95 

65 

OO 

3 ° *94 

90 

OO 

00.00 

15 

3 ° 

4583.92 

40 

30 

34*42 

65 

30 

1993.04 




l6 

OO 

72.94 

4 i 

OO 

07.62 

66 

OO 

• 54*97 




l6 

30 

61.61 

4 i 

30 

3580.54 

66 

30 

16.75 




17 

00 

49*94 

42 

00 

53*17 

67 

OO 

1878.37 




17 

30 

37*93 

42 

30 

25*54 

67 

30 

39*84 




18 

OO 

25*59 

43 

OO 

3497.62 

68 

00 

1801.16 




18 

30 

12.90 

43 

30 

69.44 

68 

30 

1762.33 




19 

OO 

4499.87 

44 

OO 

40.98 

69 

OO 

23*36 




19 

3 ° 

86.51 

44 

30 

12.26 

69 

30 

1684.24 




20 

OO 

72.81 

45 

OO 

3383*27 

70 

OO 

45.00 




20 

30 

58.78 

45 

30 

3354*01 

70 

30 

05.62 




21 

OO 

44.41 

46 

OO 

24.49 

7 1 

OO 

1566.10 




21 

3° 

29.71 

46 

3° 

3204.71 

7 1 

30 

26.46 




22 

OO 

14.67 

47 

OO 

64.68 

72 

OO 

1486.70 




22 

30 

4399*30 

47 

30 

34*39 

72 

3° 

46.81 




23 

00 

83.60 

48 

OO 

03.84 

73 

OO 

06.81 




23 

30 

67*57 

48 

30 

3 J 73*°4 

73 

30 

1366.69 




24 

OO 

51.21 

49 

OO 

4 1 *99 

74 

OO 

26.46 




24 

30 

34*52 

49 

3° 

10.69 

74 

30 

1286.12 




25 

OO 

17*51 

50 

OO 

3 ° 79* T 5 

75 

OO 

45.68 






































446 


MA P CONS TR UCTION. 


utes, are then reduced to minutes and their decimals. These 
are multiplied by the corresponding distances, taken directly 
from Table XXIII or Tables XXV and XXVI, and corre¬ 
sponding to one minute for the map scale selected. 

Having found these quantities, they are platted as differ¬ 
entials of latitude northward from the last latitude line ruled 
on the projection, and as differentials of longitude westward 
from the last longitude line platted on the projection, and 
perpendiculars are erected, the intersections giving the exact 
position of the point. When all the points falling within the 
area of the map have been platted in this manner, the accu¬ 
racy of the plat may be tested by computing the differences 
of latitude and longitude backward by subtracting the min¬ 
utes and fractions from the next greatest ten-minute projec¬ 
tion line. They are also to be checked by measuring the 
distance between the various points as given in the computed 
geodetic coordinates and reduced to the map scale. 

189. Scale Equivalents. —The proper scale to employ 
for a topographic map is dependent wholly upon the pur¬ 
poses to which that map is to be put. Where it is desirable 
to show the topography of a large area of country on a single 
map, the smallest scale should be employed which will permit 
of representing the features it is desired to show, for the 
reason that the smaller the scale the larger the area brought 
in view of the eye on one piece of paper. Again, the scale is 
affected materially by the method of representing relief. If 
contours are employed, such a scale must be used as will 
permit of all the contours being shown in the proper places 
without crowding them too closely upon the map, on the one 
hand, and yet without their being so far apart, on the other 
hand, as to detract from the expression which they give to 
the surface relief. In general it may be stated that for a 
given contour interval the smallest scale should be chosen 
which will permit of properly platting the contours, for thus, 
by bringing them closer together, the best effect is obtained 


SCALE EQUIVALENTS. 447 

in depicting the forms mapped, and the largest area is shown 
on a given map sheet. 

For exploratory maps scales as small as one to one- 
millionth may be employed. For geographic maps scales of 
1 : 63,360 to 1 : 500,000 may be most satisfactorily employed. 
For general topographic maps scales of 1 : 10,000 to 1 : 63,360 
will be sufficiently large to permit of properly representing the 
terrane. For cadastral maps scales of 1:2,500 to 1:10,000 
may be used, and for these as well as for detailed topographic 
maps for the working out of engineering problems scales as 
large even as 10c or 200 feet to the inch may be employed. 

Table XXVIII gives in fractional form the ratio of inches 
corresponding to a given distance in feet, miles, meters, or 
kilometers, as represented by the reduction from nature 


Table XXVIII. 

SCALE EQUIVALENTS FOR VARIOUS RATIOS. 


Feet 

to One Inch. 

Miles 

to One Inch. 

Meters 
to One Inch. 

Kilometers 
to One Inch. 

Ratio 

(Number Inches). 

IOO 

0.019 

30.480 

0.030 

I 

1,200 

400 

0.076 

121.920 

0.122 

I 

4,800 

500 

O.095 

152.400 

0.153 

I 

6,000 

800 

O.151 

243.840 

O. 244 

I 

9,6co 

$ 33 * 

O. 158 

254.000 

O.254 

I 

10,000 

1,000 

0.189 

304.801 

0-305 

I 

12,000 

2,500 

0-473 

762 .0 

O.762 

I 

30,000 

2,640 

0-5 

804.6 

0.805 

I 

31,680 

3.333* 

0.631 

I,Ol6 .0 

I .016 

I 

40,000 

3.750 

O.710 

IU 43 

I -143 

I 

45,000 

5,000 

0-947 

1,524.0 

I.524 

I 

60,000 

5,208 

0.988 

1,587 

1.587 

I 

62,500 

5,280 

I 

1,609.2 

1.609 

I 

63,360 

6,666| 

I.263 

2,032.0 

2.032 

I 

80,000 

7 . 5 oo 

1.42 

2,286.0 

2.286 

I 

90,000 

8 , 333 * 

1-578 

2,540.0 

2-54 

I 

100,000 

10,416 

1.976 

3 , 174-9 

3-175 

I 

125,000 

10,560 

2 

3,218.4 

3.218 

I 

126,720 

i6,666| 

3 -I 56 

5,080.0 

5-08 

. I 

200,000 

20,832 

3-952 

6,349.8 

6-35 

I 

250,000 

21,120 

4 

6,436.8 

6-437 

I 

2 53,440 

31,680 

6 

9 , 655 - 2 

9.656 

I 

380,160 

41,666! 

7.891 

12,699.6 

12.7 

I 

500,000 

83 , 333 * 

I 5-783 

25,400.0 

25-4 

I 

1,000,000 






















448 


MAP CONSTRUCTION. . 


to maps of different scales. This table gives a number of 
those equivalents corresponding to the more usual map scales 
employed both in engineering topography and in the topo¬ 
graphic and geographic atlases published by State and Gov¬ 
ernment organizations. 

Table XXIX gives equivalent ratios showing the num¬ 
ber of inches corresponding to one mile. 

Table XXIX. 

RATIOS EQUIVALENT TO INCHES TO ONE MILE. 


i inch 

to 

1 

mile. 

Equivalent 

1 

63,360 

inches 

< < 

1 

< < 

< < 

1 

50,688 

“ 

«1 

1 

a 

<« 

1 

47,520 

1* “ 

(1 

1 

• •• 

1 « 

1 

42,240 

if « 

< < 

1 

* i 

1 < 

1 

39,600 

if 

< t 

1 

i i 

1 1 

1 

38,016 

2 “ 

< < 

1 

1 < 

1 « 

1 

3U68o 

2 \ 

< < 

1 

< t 

«t 

1 

25,344 

3 

< < 

1 

«< 

1 < 

1 

2 1,120 

4 

< i 

1 

i ( 

< < 

1 

15,840 

5 


1 

< * 

1 ( 

1 

12,672 

6 

< 1 

1 

«c 

<« 

1 

10,560 


CONVENIENT EQUIVALENTS. 

i inch = 2.54 centimeters. 

1 foot = 0.3048 meter. 

1 yard = 0.9144 meter. 

1 mile = 1.60935 kilometers. 

1 square yard = 0.836 square meter. 

1 acre = 0.4047 hectare. 

1 square mile = 259 hectares. 

1 square mile = 2.59 square kilometers. 

1 cubic foot = 0.0283 cubic meter. 

1 cubic yard = o. 7646 cubic meter. 

1 acre = 209 feet square, nearly. 

1 acre = 43,560 square feet = 4840 square yards. 

1 statute mile = 1760 yards = 5280 feet = 63,360 inches. 
1 cubic foot = 7.48 gallons = 0.804 bushel. 

1 cubic foot of water weighs 62.4 pounds. 

1 wine gallon = 8.34 pounds water. 

1 wine gallon = 231 cubic inches. 

1 avoirdupois pound = 7000 grains. 

1 troy pound = 5760 grains. 







CHAPTER XX. 


TOPOGRAPHIC DRAWING. 

190. Methods of Map Construction. —There are two 
general modes of representing artificially in map form the 
various topographic features surveyed. These are: 

1. Representation on paper by means of various conven¬ 
tional signs used in topographic drawing; and 

2. Representation of the relief in a miniature model, 
special features of culture or drainage being denoted by 
conventional signs painted thereon. 

A third method, and one which is most graphic in the 
depiction of surface forms, is by making a photograph of a 
model map, the result being a relief map. 

The various processes employed in indicating the results 
of a topographic survey on paper are described as topographic 
drawing (Art. 191). Those employed in representing the 
same on a model map are known under the general expression 
modeling (Art. 198). 

Relief maps are photomechanical copies of model maps 
(Art. 198). 

191. Topographic Drawing. —In drawing topographic or 
geographic maps, a few conventional signs have been gener¬ 
ally accepted for the representation of the various features of 
the terrane. Wavy lines, corresponding in plan to their posi¬ 
tions upon the surface of the earth, are employed to represent 
outlines of seacoast or lakes, margins of streams, etc. In 
representing a stream it is customary to begin at the head¬ 
waters, where the stream is smallest, with the finest possible 

449 


450 


TOPOGRAPHIC DRAWING. 


line, increasing its width as the stream increases in size, 
until toward the mouth, if the map scale will permit it—a 
single line failing to be sufficient for its representation—it 
becomes necessary to double-line it, the two lines represent¬ 
ing as nearly as possible to scale the outlines of the shores of 
the stream. Water surfaces, such as those of oceans or lakes 
or of broad rivers, are indicated by parallel lines called water 
lines, somewhat like contour lines, and the distance between 
them at the shore is about equivalent to the width of the 
line, this distance being increased rapidly away from the 
shore until the lines disappear entirely. (Figs. 43 and 146.) 

Surface forms of relief are represented by two general 
systems: 

1. The vertical system, by hachures (Figs. 19, 141, and 
143), which are short lines parallel to the direction of flow of 
water on the slope and based upon a scale of shades so 
graduated as to represent the relative amount of light which 
may be reflected from various degrees of slope; and 

2. The horizontal system, by contours (Figs. 4, 135, and 
1 39), which are continuous lines representing equal vertical 
intervals and which are in fact projected plans of the line at 
which a water surface (of the ocean, for instance) would inter¬ 
sect the surface of the earth were it raised successively by equal 
amounts. These contours are, then, curved lines which repre¬ 
sent in plan the country as it would appear if it were cut by a 
series of equidistant horizontal planes. The contour interval , 
as it is called, or the distance between two contour lines, may be 
assumed at pleasure, but must be constant for the same map. 

Still another method of representing surface slopes is by 
crayon or brush shading (Fig. 138), so as to give the effect 
produced by hachures, but in a uniform tint; and still another 
and perhaps the best method of all is that of shading a 
contour map in such manner as to produce the graphic relief 
effect resulting from hachures, while at the same time it retains 
the quantitative property given by contours. (Fig. 136.) 


METHODS OF MAP CONSTRUCTION. 


451 



Fig. 135.— Contour (D), Shade-line (£), and Hachure Construction (A). 













































































452 


TOPOGRAPHIC DRA WING. 


The representation of relief by hachures is graphic only . 
By this method quality of relief is the first consideration, and 
quantity or relative amount of relief is secondary. (Figs. 135 
and 143.) Where quantitative relief is necessary, as in the 
work of the engineer or geologist, a contour map is essential. 
While such a map is possessed of mathematical qualities and 
clearness that are lacking in the hachured map, it fails to a 
large degree in representing the details of the surface, and, 
moreover, its representation of surface forms is difficult of 
interpretation by the inexpert. The representation of relief 
by hachures has been characterized as a graphic system with a 
conventional element, and the contour method of representing 
relief as a conventional system with a graphic element, the 
latter being obtained when the contour interval is so small as. 
to produce a shading in the map, as when the scale selected 
is relatively small. (Figs. 4, 34, and 35.) 

In any form of map-shading the lighting may be taken 
from one of two directions. If vertical, that is, from above, 
no high lights are introduced, but the highest summits have 
the lightest tint. The better and more commonly accepted 
method of shading is to assume that the light comes from an 
angle of 45 0 to the left, or, in other words, from the upper 
left-hand corner of the map; the northwest corner (Figs. 136, 
138, and 143) and the high lights are, therefore, those which 
are tangent to this direction. 

Two effective methods of representing relief which are 
most useful in sketching in the field on a plane-table board 
are: 

1. By means of sketch or broken contours; and 

2. By means of crayon shading. 

Sketch contours have the general form of continuous con¬ 
tour lines and represent the slopes in plan pictorially. They 
also give differences of altitude relatively, but the quantity 
of relief is not accurate over any great space on the map, 
(Figs. 15, 23, and 137)- When the final drawing is made in 



Fig 136. —Shaded Contour 






























') 



















I 

1 






















































CONTOUR LINES. 


455 


office from such a sketch map, the altitudes which have been 
determined everywhere give points upon which connected 
contour lines can be constructed by following the forms in¬ 
dicated by the sketch contours. 

Crayon or brush shading bears about the same relation to 
hachure drawing as do sketch contours to continuous contour 



Fig. 137. —Sketch Contours. Xalapa, Mexico. 

lines. By the means of a soft crayon or pencil the shapes 
and steepness of slope of the terrane can rapidly be repre¬ 
sented in the field, and, if it is desired, the same can afterwards 
be worked up into a finished hachure map, or, providing ele¬ 
vations are sufficiently abundant, into a contour map. 

192. Contour Lines. —In order to represent the terrane 
quantitatively, that is, to show not only the slopes and shapes 
of the country and the relative steepness of the hills, but 





456 


TOPOGRAPHIC DRAWING. 


also amounts and differences in elevation at any point, a 
system of map construction is employed called contouring. 
Contour lines are lines of equal elevation above some datum 
* as the mean sea-level. They are drawn at regular vertical 
intervals, their distances apart being dependent upon the 
horizontal scale of the map, and they thus indicate not only 
actual differences of elevation, but degrees of steepness or 
grades. 

Contour lines express three degrees of relief: 

1. Elevation; 

2. Horizontal form; 

3. Degree of slope. 

The manner in which they express these is illustrated in 
the following perspective view and contour sketch (Fig. 139), 
taken from the explanatory text accompanying the atlases of 
the U. S. Geological Survey. 

The sketch represents a valley between two hills. In the 
foreground is the sea with a bay which is partly closed by a 
hooked sand-bar. On either side of the valley is a terrace; 
from that on the right a hill rises gradually with rounded 
forms, whereas from that on the left the ground ascends 
steeply to a precipice which presents sharp corners. The 
western slope of the higher hill contrasts with the eastern by 
its gentle descent. In the map each of these features is 
indicated, directly beneath its position in the sketch, by 
contours. The following explanation may make clearer the 
manner in which contours delineate height, form, and slope: 

1. A contour indicates approximately height above sea- 
level: in this illustration the contour interval is 50 feet; 
therefore the contours occur at 50, 100, 150, 200 feet, and so 
on, above sea-level. Along the 250-foot contour lie all 
points of the surface 250 feet above sea; and so with any 
other contour. In the space between any two contours occur 
all elevations above the lower and below the higher contour. 
Thus the contour at 150 feet falls just below the edge of the 


CO A 1 O UK SKK1 CH1NG 


457 



Fig. 138 . —Relief by Crayon Shading 















































































CONTOUR CONSTRUCTION. 


459 


terrace, while that at 200 feet lies above the terrace; there¬ 
fore all points on the terrace are shown to be more than 150 
but less than 200 feet above sea. The summit of the higher 
hill is stated to be 670 feet above sea; accordingly the con¬ 
tour at 650 feet surrounds it. In this illustration nearly all 



Fig. 139.—Contour Sketch. 

the contours are numbered. Where this is not possible, 
certain contours, as every fifth, are made heavy and are num¬ 
bered ; the heights of others may then be ascertained by 
counting up or down from a numbered contour. 

2. Contours define the horizontal forms of slopes: since 
contours are continuous horizontal lines conforming to the 
surface of the ground, they wind about the surfaces, recede 
into all re-entrant angles of ravines, and define all promi- 
























460 


TOPOGRA PH/C D RA WING. 


nences. The relations of contour characters to forms of the 
landscape can be traced in the sketch and map. 

3. Contours show the approximate grade of any slope: 
the vertical space between two contours is the same whether 
they lie along a cliff or on a gentle slope; but to rise a given 
height on a gentle slope one must go farther than on a steep 
slope. Therefore contours are far apart on the gentle slopes, 
and near together on steep ones. 

193. Contour Construction —In representing on a map 
the relief or'configuration of the surface of the land by means 
of contour lines two objects are to be kept constantly in 
mind. 

1. The contour lines must be so constructed as to always 
maintain with accuracy relative and absolute elevations. 

2. They must be so drawn as to give a distinct picture of 
the shapes of the country as though viewed from a consider¬ 
able altitude above the surface. 

Contour lines are actual mathematical horizontal projec¬ 
tions, to a given scale, of the intersection of the surface of the 
terrane by imaginary horizontal planes. Moreover, these 
imaginary planes are, for any given map, accepted as being at 
equal and uniform vertical distances apart. 

Contour lines are drawn during the processes of eye 
“sketching” (Fig. 139) by making an imaginary projection 
in plan of numerous sections through the hill viewed. This 
is illustrated in Fig. 140, which represents a section through 
a hill and indicates graphically the manner in which the con¬ 
tours are projected. Each individual contour line is the pro¬ 
jection of the intersection of the horizontal plane through the 
hill. In learning to sketch contours the topographer will do 
well at first to keep in mind clearly this form of construction, 
and wherever in doubt as to the mode of representing any 
feature he should construct a profile sketch of it, draw hori¬ 
zontal section lines through it and project them in plan. 
In this manner he will soon learn to mechanically perform 


HA CH URE CONS 1R ULT1 ON. 


461 

this operation in imagination, and later, as his skill develops, 
will draw contour lines without performing any intermediate 
mental operations. 



Fig. 140.—Contour Projection. 


194. Hachures. —Hachuring is a conventional method of 
representing the relief of a country by shading, in short dis¬ 
connected lines, in such manner as to indicate its slopes and 
relative steepness. The distance apart of the lines, their 
weight or thickness, and accordingly the degree of density 
which they produce on the map give qualitative and not 
quantitative results. These lines are of varying lengths, and 
are drawn in the direction of the slopes and in such manner 
as to horizontally encircle them in bands, and the width of 
these is intended to represent equal vertical heights. 

In order, therefore, to construct a hachure map it is neces¬ 
sary to sketch approximate contour lines (Fig. 141), the dis¬ 
tance between any two of these representing approximately a 
fixed vertical distance. Between these contour lines and at 
right angles to them are drawn the hachure lines, the contours 
being only penciled in and the hachures inked so that ultimate¬ 
ly the contour lines disappear. The hachure lines, as already 
stated, are not made continuous, but rest on the horizontal 

























462 


TOPOGRAPHIC DRA WING. 


or contour curves, which are thus indicated by the termini of 
the hachure lines. In haciHiring a map the following general 
directions are suggested: 

Commence with the lighter slopes in the lightest line, in 
order that the intensity of the tint may be increased with 
more regularity. (Fig. 141.) When the projections of the 



Fig. 141.—Hachure Construction. 


horizontal sections or contours are parallel the hachures are 
at right lines normal to both curves, but when they are not 
parallel the hachures radiate, their extremities being respec¬ 
tively normal to the curves at which they terminate. Hachures 
are in sections or bands which should not be continuous with 
the adjoining ones, but should terminate in the spaces be¬ 
tween them, thus accentuating the contour lines without ink¬ 
ing them. When the slope suddenly becomes abrupt the 
tint must be deepened by increasing the width of the hachure 

t 

near the extremities or by interpolating short lines between 
the original hachures. Hachures are made shorter and wider 
for steep slopes, and are lenghtened and narrowed as the in¬ 
clination decreases. 
















CONVENTIONAL SIGNS 


463 


The first principle upon which hachures are constructed is 
that the steeper the slope the less light is received in the in¬ 
verse ratio of its length. Various methods of expressing the 
degree of shade, or the ratio of black to white , have been 
adopted by various draftsmen. The Enthoffer or American 
method is to indicate the degree of slope by varying the dis¬ 
tance between the hachure lines, the distance between the 
centers of lines to be .02 of an inch plus one-fourth of the 
denominator of the fraction denoting the declivity, expressed 
in hundredths of an inch. The lines are accordingly made 
heavier as the slope is steeper, and finer for gentle slopes, in¬ 
creasing in width until the blank spaces between them equal 
one-half the breadth of the lines. (Fig. 142.) The German 



Fig. 142.—Shaded Hachures. 

or Lehmann’s method consists in using nine widths of lines 
for slopes from zero to 45°, the first being white and the last 
black. For the intermediate slopes the proportion of white 
to black is as 45 0 minus the angle of slope is to the angle 
of slope. Steeper slopes than 45 0 are represented by short, 
heavy lines, parallel to the contour lines. 

195. Conventional Signs.—Various conventional signs 
are employed in topographic drafting to represent roads, 
houses, woods, marshes, the shapes of hills, etc. These 
signs may be divided into three general classes : 

1. Signs to represent culture or the works of man. 






















































































464 


TOPOGRAPHIC DRA WING. 


2. Sign^ to represent hydrography or water. 

3. Signs to represent hypsography or relief. 

In the making of a geographic map or of a topographic 
map for the use of a government or State, only such culture 
should be represented as is of a permanent or public nature. 
This includes all highways, bridges, railways, political bound¬ 
ary lines, and houses. (Figs. 144 and 145.) Though the 
latter are not of a public nature, yet they are comparatively 
permanent and are prominent topographic features. 

For purposes of legibility in deciphering a map it is 
desired to use various colors in representing different features, 
and the color scheme selected by the U. S. Geological Sur¬ 
vey, which is one of the best, employs ^1) black for all culture 
and lettering; (2) blue for hydrography; and (3) brozvn for 
surface relief. 

In the representation of hydrography , or water forms, 
conventional signs must be adopted (Fig. 146) for streams, 
lakes, marshes, canals, glaciers, etc. 

For the representation of hypsography , or surface relief, 
conventional signs must be adopted (Fig. 147) for the repre¬ 
sentation of slopes, by means of contours or hachures, with 
separate symbols to indicate depressions of the surface, also 
sand-dunes, cliffs, etc. 

In addition to the above conventional signs used in depict¬ 
ing public culture and the more usual topographic forms, a 
great variety of signs are used to represent minor forms, as 
lighthouses, mines, quarries, churches, different kinds of 
houses, as barns, private residences, mills, also to represent 
different kinds of woods and cultivated fields. These are 
described and illustrated in various works on topographic 
drawing. The only one of these of importance which may 
be further characterized here are woods, and for these conven¬ 
tional signs may be adopted to indicate the wooded character 
of the country, or, better, a light green tint may be washed 
over the wooded portion of the map. 


HACHURED AND CONTOURED HILL—VARIOUS SCALES . 465 



Hachure. 


Spale, 1 inch - 4 miles. 



Con tour 
200 a. 



interval 
500 a 




Fig. 143.— Hachured and Contoured Hill on Different Scales. 


(After S. Enthoffer.) 












































































































































' 

















Fig. i45* Conventional Signsj Miscellaneous Symbols and Boundary Lines 












































- 


, 

. 


■ 













































. 















Fig. 146.—Conventional Signs ; Hydrography. 

































































































Contour' System 


Ldepression Contour vs- 



Sand 


Sand Dunes 


VvJ&'.v V.v 

■ w 'wiviv’Xi\wW* JT 



Levels 



s->,.IIIIIII' 1 ,% 

S«'"" 


CUCfs IHcichured J 
Mine Dumps 



Fig. 147.—Conventional Signs ; Relief or Hypsograph* 

473 







N 




/ 


\ 




> 


/ 


: 










( 







i 










/ 



* 


/ 


t 

>. 



/ 










r'lG. 148. —Conventional Signs; Lettering. 



7 


























D RA FTJNG INS TR UMEN TS , 


477 


196. Lettering. —As with conventional signs, so with 
lettering. Various books are published describing the mode 
of constructing different kinds of letters. It is desirable 
here, therefore, only to give a general outline of the princi¬ 
ples on which topographic lettering should be executed. 
Letters used in describing different topographic forms may be 
divided into four corresponding classes, and there should be, 
therefore, as many different styles of letters. Those preferred 
by the author and shown in Fig. 148 consist of— 

1. Roman letters of various sizes for the names of civil 
and political divisions, as cities, States, etc. 

2. Italic and script letters of various sizes for the names 
of hydrographic features, as lakes, rivers, etc. 

3. Vertical block of various sizes to represent hypsographic 
features, as mountains, plateaus, valleys, etc. 

4. Slanting block to represent public work, as railroads, 
trails, etc. 

197. Drafting Instruments —It is not deemed desirable 
to describe in detail the various instruments commonly used 
in topographic drawing. These can be found fully described 
in catalogues of instrument-makers and in works on mechanical 
and topographical drawing, lettering, etc. A few instruments, 
however, which are less common and which are peculiarly 
serviceable to the topographic draftsman will be briefly 
characterized. 

The pantograph is a parallel linked-motion apparatus for 
enlargement or reduction of maps. It is of occasional assist¬ 
ance in the reduction or enlargement of compiled map material, 
and is constructed on the theory of parallelograms. (Fig. 149.) 
The pantograph is, however, a comparatively inaccurate 
instrument because of the great play between the various 
parts. If accuracy is attempted, none but the most expensive 
and heavily constructed instruments should be used. There 
is an inconvenient variety of combinations in the adjustment 
and use of this instrument. The essentials are that the fixed, 


478 


TOPOGRAPHIC DRA WING. 


the tracing, and the copying points shall lie in a straight line 
on at least three sides of the jointed parallelogram. 

Two very useful instruments to the topographer are 
proportional dividers and three-legged dividers , the first of 
which is very serviceable in the reduction or enlargement of 
small portions of maps, and the second in transferring work 



from one map to another. In this operation two of the legs 
are set on fixed points common to both maps, as the inter¬ 
sections of projection lines, and the third is used as a pointer 
to transfer the position desired. This instrument is especially 
useful in the transferring and adjustment of lines from the 
traverse sheets (Fig. 2) to the sketch sheets on which they 
are to be adjusted to the triangulation positions (Fi g- 3 )- 

For the construction of projections the topographer needs 
a first-class beam compass and an accurately graduated steel 
scale. The ordinary triangular boxwood scale is well gradu¬ 
ated and is useful in the projection of very small scale-maps; 
but for larger ones a long steel scale, preferably divided to 
the scale of the map work, will give much more satisfactory 
results. 

The use of vernier protractors is fully described in Article 89, 
while plane-table paper and like accessories are discussed in 
Chapter VIII. 

198. Model and Relief Maps. —Relief maps are of two 
general kinds: 

I. The model, which is not a map in that it has three 




MODEL AND RELIEF MAPS. 479 

dimensions, is bulky, and cannot be inserted in atlases or 
books; and 

2. The reproduction of the model by some photo-mechan¬ 
ical process which results in a print in map form of the model. 

Models have certain striking advantages over maps of all 
kinds, because they represent graphically the surface relief in 
a manner superior to that of hachure or shaded contour maps, 
besides which they represent quantitatively the relative relief 
in a more simple and legible manner than do contour maps. 

There are two general varieties of models: 

1. Those in which surface slopes are smoothed out in such 
manner as to practically represent the surface of the country 
as it appears in nature, and which, while possessing inertly 
relative relief, lack the element of absolutely quantitative relief 
possessed by contour maps; and 

2. Those in which the slopes are represented by steps, 
each of which is a contour interval apart on some scale; and 
while this does not imitate nature as exactly as the first form, 
it possesses an absolute quantitative element which makes it 
superior for many purposes. 

Models are of especial value for educational purposes , in 
teaching those who are not familiar with maps something of 
geography which they would not appreciate by looking at the 
flat surface of a map. They are nature in miniature. In 
addition they have great economic value to the mining engi¬ 
neer and the geologist: to the former in obtaining a true 
appreciation of the differences in level and direction of the 
numerous shafts and tunnels which permeate the ground in 
the mining districts; to the latter because many important 
structural features and relations are presented to the eye at a 
glance, and because both the surface topography and its rela¬ 
tion to the underground topography are brought together in 
their proper relationship. For exhibition purposes they are 
unsurpassed in that they possess the quality, next best to that 
of moving objects, which catches the attention of the beholder 


480 


TOPOGRAPHIC DRA WING. 


and attracts him to a further study of the subject represented 
in a way which no map can. 

Relief maps possess numerous advantages over hachure or 
contour maps chiefly because they give a more graphic idea 
of the surface relief than can be had from any artificial method 
of map construction. They are made by photographing a 
model which is set up in such manner as to get the proper 
lighting, that which will bring out lights and shadows most 
effectively. For the successful reproduction of a relief map 
it is essential that the model should not be colored and its 
surface should be dull, not glossy; there should be a slight 
yellow tint in the material composing it, the effect of which 
is to produce a smoother, more subdued lighting and shading, 
and to do away with the glaring high lights coming from a 
white model. (Fig. 150*) Relief maps can be given certain 
quantitative values if they are reproduced from contour 
models (Fig. 151), and they may be given certain further 
values by simple lettering in black. 

The groundwork for the construction of a model is a good 
contour map, in addition to which the modeler should possess 
a fair knowledge of the topography of the country, obtained 
by personal inspection, and he should have at hand good 
hachured maps and photographs which will aid him in inter¬ 
preting the topographic forms. It is the personal expres¬ 
sion that is brought into a model, by the appreciation of 
the country obtained from a knowledge of it, which results 
in the difference between a good and a bad model of the 
same region as produced by two modelers from the same data. 
The treatment of the vertical or relief element required to 
represent the individuality of a given district is especially im¬ 
portant. 

199. Modeling the Map. —The amount of relief to be 
given a model, that is, the amount of exaggeration in vertical 
scale as compared with the horizontal, is a question of great 
importance. The tendency is always to exaggerate the ver- 


MODEL AND RELIEF MAPS. 



481 


Fig. 150.—Relief Map from Catskill Model. 

Vertical and horizontal scales, both 1 inch to 1 mile. Modeled by E. E. Howell. 




















V 




MODELING THE MAP. 


483 


tical element too much, the result of which is to produce a 
false effect by diminishing the proportionate width of valleys, 
thus making the country seem more rugged and mountainous 
than it is. Another effect is to make the area of the region 
represented appear small, all idea of the extent of the country 
being lost. Messrs. E. E. Howell and Cosmos Mindeleff, of 
Washington, D. C., two of the most expert model-map- 
makers, agree that it is almost impossible under most circum¬ 
stances to use too low relative relief. Mr. Mindeleff says that 
on a scale of six inches to a mile no exaggeration at all is 
necessary, the ratio of vertical to horizontal scale being as 
1 to 1. For smaller scales than this the vertical exaggeration 
may be 2 to 1 or 3 to 1. He says further that “the absolute 
and not the relative amount of relief is the desideratum. 
For small-scale models I have found half an inch of absolute 
relief ample.” In a very handsome model of the United 
States, made by Mr. Mindeleff, a proportion of 10 to 1 was 
used, but it is believed from the appearance of the resulting 
model that 6 to 1 would have been even more satisfactory. 
Some of the most effective of recent models are made to 
natural scale, i.e., without any exaggeration of vertical scale. 
(Fig. 150.) 

For the making of model maps a number of methods have 
been employed, the majority of which are so crude or so in¬ 
ferior to the better methods as to call for scant recital here. 
One of the first employed consists in drawing cross-section 
lines at regular intervals over a contoured map, and, if the 
topography is intricate, corresponding lines at right angles. 
These sections are transferred to thin strips of cardboard or 
similar material and cut down to the surface line, thus form¬ 
ing the cross-section. These are mounted on a suitable base¬ 
board and the cavities between them filled in with plaster or 
wax or other easily worked material. The topography is then 
carved down to the form of the country as indicated by the 
upper edges of the strips. This method is crude and laborious. 


484 


TOPOGRAPHIC DRAWING. 


Where no contour map is obtainable as a base and the 
known elevations are few and scattered, one of the simplest 
methods of producing a model map is by driving pins into a 
base-board, each to a height corresponding to the elevation 
of the point it represents. The map is then built up in wax 
or moist clay by laying this on the base-board and bringing 
it up to the level of the summits of the pins, and then work¬ 
ing in the details of the map by practically sketching it in as 
a sculptor would, following a hachure or other map of the 
country as a guide. 

Another method, practically the converse of that last 
described, may be satisfactorily employed where the map 
material is scanty. A tracing of the map enlarged to the 
required size is mounted on a frame. Another but deeper 
frame, large enough to contain the mounted tracing, is made 
and laid upon a suitable base-board, upon which is mounted a 
copy of the map. Upon this base-board the model is then 
built up in clear wax, the low areas first. Horizontal control 
is obtained by pricking through the mounted tracing with a 
needle-point, and vertical control by measuring down with a 
straight-edge, sliding on the top of the deep frame. 

Model maps are sometimes made by carving or cutting 
down instead of modeling or building up , a solid block of 
plaster being used, and this being carved down so as to 
produce a series of steps similar to those made by building 
up contours. 

The best and most modern method of making map models 
is that now more generally employed by the professional 
model-makers. This consists of building up the model and 
modeling instead of carving the detail. The ratio of relief 
or vertical to horizontal scale having been determined, thin 
cardboard or wooden boards are procured of the exact thick¬ 
ness of the contour interval which the modeler proposes 
using. He then takes a contour map, enlarged or reduced, 
as the case may be, to the scale of his model and traces on 


DUPLICATING THE MODEL BY CASTING. 485 


the boards the outlines of each separate contour. Then with 
a knife, scissors or scroll-saw, following the contour line on 
the board, he cuts out each contour and lays each of these 
outline contour boards one upon the other, thus building 
them up in steps, the height of each of which bears the 
proper relation, because of the thickness of the material, to 
the vertical scale. The result is a completed model in steps. 
The re-entrant angles of the steps are then filled in with 
modeling clay or wax or some similar substance, so as to 
produce a smooth outline. 

The best material for modeling is wax; but if much 
modeling material is to be used, clay may be kept sufficiently 
moist to be worked. Some modelers find clay mixed with 
glycerine instead of water works most satisfactorily, because 
it does not dry. The filling-in process is the most important 
in the making of a model map, for in this the modeler must 
show his knowledge of and feeling for topographic forms, 
in the interpretation of not only such hachured and other 
maps as he has to guide him, but of the country, if he has 
examined it as he should. 

200. Duplicating the Model by Casting. —The model 
resulting from the above operations is practically the base 
only of the completed model map. It is the common prac 
tice to make a replica by taking from the first a mould with 
plaster of Paris, and from this a plaster cast. The common 
mistake is made of making a solid cast by filling in a frame 
which has been built around a model, the result being so 
heavy and cumbersome as to be of little use. The best 
modelers say that it is wholly unnecessary to make a cast 
which is more than | or i| inches in thickness of plaster. 
This is procured by incorporating in the plaster tow or bag¬ 
ging or netting of various kinds, the result being to make the 
cast light and strong, though the expense is slightly in¬ 
creased. Such casts can be readily and even roughly handled 
without breakage. 


486 


TOPOGRAPHIC DRA WING. 


In making the final casting from the mould the process is 
repeated. The model for the making of the mould or the 
latter for the after-process of casting should not be varnished, 
as the finer details are thus lost. The mould should be pre¬ 
pared with a solution of soap, so that nothing is left on the 
surface but a thin coat of oil, which is taken up by the plaster 
of the cast. With care and skill a cast may be thus pro¬ 
duced which is but little inferior in point of sharpness to the 
original model. 

The plaster model being completed, only such little paint¬ 
ing of names and places as may be necessary to make it 
intelligible should be done before photographing for the pro¬ 
duction of the relief map, after which it may be colored as 
desired to represent any other subject and varnished. (Fig. 
150.) 

Other materials than plaster of Paris have been used for 
making models. Some modelers, after cutting the wooden 
contours and fitting these together with wooden pegs, carve 
away the steps left by the contours with graver’s tools. 
This is an exceedingly laborious and difficult process, and 
the resulting model lacks expression and looks as wooden as 
the material from which it is made. Many efforts have been 
made to use papier mache, but owing to the distortion and 
warping in this, because of the varying degrees of moisture 
in the atmosphere and the material itself, no success has as 
yet attended its use. 

The form of model used in depicting underground work¬ 
ings in a mine is by making a skeleton model of cardboard 
and glass. A rectangular box of glass is made of such size 
to scale as to include the cubic contents to be modeled. In 
this are glued or suspended by wires, etc., painted sheets of 
cardboard at such inclinations as to graphically represent the 
various tunnels, shafts, etc., or the ore-bearing strata, as 
desired. 

A very effective form of model is made by pasting 


PAPER CONTOUR MODEL MAPS . 


487 



Fig. 151.—Relief Map from Contour Model. 

Scale 1 mile to 1 inch; contour interval 20 ft. Modeled by Wm. Stranahan. 



























PAPER CONTOUR MODEL MAPS. 


489 


together paper contours. Fig. 151 shows such a model, made 
by taking the printed sheets of the U. S. Geological Survey 
20-foot contour map of the area depicted. One sheet had 
to be taken for each contour interval, and in all 30 to 50 
sheets were used. The modeler followed carefully with 
scissors each contour line, and then superimposed each sheet 
on the next lower. By having printed paper bearing a fixed 
relation in thickness to the contour interval an exact quanti¬ 
tative reproduction of each 20-foot contour in nature is 
obtained. 


REFERENCE WORKS ON TOPOGRAPHY. 


No attempt has been made in the following list of books 
bearing on the subject of topography to include all those 
published. The endeavor has been, however, to in-clude such 
as have been consulted by the author in the preparation of 
this volume, and a few others which have a particular bearing 
upon the subject. They are enumerated here that the reader 
may know where to look for more detailed information on the 
various branches touched upon in the preceding text. 

Airy, Wilfrid. Probable Errors of Surveying by Vertical Angles. Proc. 

Inst. C. E., Vol. Cl, Pt. III. London, 1890. 

Bagnall, Gerald. Long-distance Leveling. Proc. Inst. Civ. Eng., Vol. 
CXXI, Pt. III. London, 1895. 

Baker, Ira O. Engineers’ Surveying Instruments. John Wiley & Sons, 
New York, 1895. 

-Leveling. D. Van Nostrand & Co., New York, 1884. 

Baltimore, Topographic Survey of City of. H. T. Douglas and Others. 

Wm. J. C. Dulaney, Printer, Baltimore, 1895. 

Board of Ordnance and Fortifications, Fifth Report. “ Range-finders." 

Government Printing Office, Washington, D. C., 1895. 

Bohn, Dr. C. Landmessung. Julius Springer, Berlin - , 1886. 

Bowhill, Maj. }. H. Military Topography. Questions and Answers 
with Volumes of Plans. Macmillan Co., New York, 1898. 

Branner, John C. Geology and Its Relations to Topography. Proc. 

Am. Soc. C. E., Vol. XXIII, No. 8. New York, October, 1887. 
Brough, Bennett H. Tacheometry, or Rapid Surveying. Proc. Inst. 

C.E., Vol. XCI, Pt. I. London, 1888. 

Carpenter, Wm. DeYeaux. Geographical Surveying. D. Van Nostrand 
& Co., New York, 1878. 

Colby, B. H. Recent Survey of St. Louis. Journal of Association of 
Engineering Societies, Vol. II, No. 3. Chicago, January, 1893. 

490 



REFERENCE IV OR NS ON TOPOGRAPHY. 


491 


Comstock, Lt.-Col. C. B. Report on Primary Triangulation of U. S. 
Lake Survey. Prof. Papers, Corps of Engineers, No. 24. Wash¬ 
ington, D. C., 1882. 

Deville, E. Photographic Surveying. Dominion Land Survey. Ottawa, 
Canada, 1895. 

Diaz, Augustin. Catalogo de Los Objectos. Commission Geographico 
Exploradora, Republica Mexicana. Xalapa-Enriquez, Mexico, 1893. 
Enthoffer, J. Manual of Topography and Text-book of Topographic 
Drawing. D. Appleton & Co., New York, 1871. 

Flemer, J. A. Photography as Applied to Surveying. Appendix 3, 
Report for 1893, U. S. Coast and Geodetic Survey, Washington, 
D. C. 

—— Photographic Methods and Instruments. Appendix 10, Report for 
1897, U. S. Coast and Geodetic Survey, Washington, D. C. 

Frome, Lieut.-Genl. Surveying. Lockwood & Co., London, 1873. 
Gannett, Henry. Manual of Topographic Methods. Monograph XXII, 
U. S. Geological Survey. Washington, D. C., 189-. 

Gelbcke, F. A. Surveys for Railway Location. Trans. Am. Soc. C. E., 
Vol. XXIX. New York, August, 1893. 

Gilbert, G. K. New Method of Measuring Heights by Means of 
Barometer. Annual Report, Director U. S. Geological Survey. 
Washington, D. C., 1882. 

Gillespie, Wm. Higher Surveying. D. Appleton & Co., New York, 

1883. 

-- Treatise on Leveling, Topography, and Higher Surveying. 

D.-Appleton & Co., New York, 1883. 

Gribble, Theo. Graham. Preliminary Surveys and Estimates. Long¬ 
mans, Green & Co., London and New York, 1891. 

Guyot, A. Smithsonian Miscellaneous Collections, No. 13. Washington, 
D. C., 189- 

Hardy, A. S.; McMaster, Jno. B.; Walling, Henry F.; Specht, George G. 
Topographical Surveying. D. Van Nostrand & Co., New York, 

1884. 

Haupt, Lewis M. The Topographer, His Instruments and Methods. 

Henry Cary Baird & Co., Philadelphia, Pa., 18S3. 

Hergesheimer, E. A Treatise on the Plane-table. Appendix 13, Re¬ 
port for 1880, U. S. Coast and Geodetic Survey. Washington, 
D. C. Revised by D. B. Wainright, Appendix 7, 1905. 

-Standard Topographic Drawings. Appendix 11, Report for 1879, 

U. S. Coast and Geodetic Survey. Washington, D. C., 1881. 
Hilgard, J. E. Polyconic Projection of Maps. Table IV, Appendix 6, 
Report for 1884, U. S. Coast and Geodetic Survey. Washington, 

D. C. 





492 


TOPOGRAPHIC DRAWING. 


Hills, Capt, E. H. Determination of Terrestrial Longitudes by Photog- 
raphy. Monthly Notices Royal Astronomical Society. London, 
Jan. 1893. 

Hodgkins, W. C., (translation of Licka, J. L.) Treatise on Wagner 
Tachymeter and Tachygraphometer. Appendix 16, Report U. S. 
Coast and Geodetic Survey for 1891. Government Printing Office, 
Washington, D. C., 1892. 

Jacoby, Henry S. Plain Lettering. Engineering News Publishing 
Co., New York, 1897. 

Johnson, J. B. Theory and Practice of Surveying. John Wiley & 
Sons, New York, 1894. 

Larned, Col. C. W., U. S. A. Military Graphics. Journal Military Ser¬ 
vice Inst., Vol. XXIII, Nos. 125 and 126. Governor’s Island, N. Y„ 
1903. 

Lee, Thomas J. Tables and Formulae Useful in Surveying, Geodesy, 
and Practical Astronomy. Professional Papers. Corps of Engi¬ 
neers, U. S. Army. Government Printing Office, Washington, 
D. C., 1873. 

Lefebvre, M. Military Landscape Sketching. Translated by Capt. W. 
W. Judson, U. S. Army. Washington Barracks, Washington, D. C., 
1902. 

Licka, J. L., translated by Hodgkins, W. C. Treatise on Wagner 
Tachymeter and Tachygraphometer. Appendix XVI, Report U. S.. 
Coast and Geodetic Survey for 1891. Government Printing Office, 
Washington, D. C., 1892. 

Lippincott, J. B. Topographic Methods. Annual of Am. Soc. Irriga¬ 
tion Engrs., Denver, Colo., 1893. 

McMaster, Jno. B.; Walling, Henry F.; Specht, George G.; Hardy, A. S.. 
Topographical Surveying. D. Van Nostrand & Co., New York, 
1884. 

Mendell, G. H. Treatise on Military Surveying. Van Nostrand & Co., 
New York and London, 1864. 

Middleton, Reginald E. Practical Observations in Tacheometry. Proc. 
Inst. Civ. Eng., Vol. CXVI, Pt. II. London, 1894. 

Mindeleff, Cosmos. Topographic Models. National Geographic Maga¬ 
zine, Vol. I, No. 3. Washington, D. C., 1888. 

Morrison, G. James Maps, Their Uses and Construction. Edward 
Stanford, London, 1902. 

Morales y Ramirez, D. Jose Pilar. Dibujo Topografico-Catastral. 
Madrid, Spain, 1874. 

Noe, D. de la, and de Margerie, E. Formes du Terrain. Imprimerie 
National, Paris, 1888. 

Ockerson, J. A., and Teeple, Jared. Tables showing Differences in 
Level in Feet, for Horizontal Reduction with Stadia. Engineer 


REFERENCE WORKS ON TOPOGRAPHY. 


493 


Corps, U. S. Army. Government Printing Office, Washington, 
D. C., 1879. 

Ogden, Herbert G. Topographic Surveys. Transactions Am. Soc. C. E., 
Vol. XXX, October, 1893. New York. 

Ordnance Survey of Great Britain. Contouring and Hill Delineation. 
London, 1854. 

Patterson, Lieut.-Col. Wm. Military Surveying and Reconnaissance. 

Triibner & Co.. Ludgate Hill, London, 1882. 

Pelletan, A. Traite de Topographie. Baudrey et Cie., Paris, 1893. 
Pierce, Josiah. Economic Use of the Plane-table. Proceedings Institute 
of Civil Engineers, Vol. XCII, Pt. II. London, 1888. 

Powell, John W. Organization of Scientific Work of General Govern¬ 
ment. Testimony before Joint Commission. Miscellaneous Docu¬ 
ment 82, U. S. Senate, Forty-ninth Congress, First Session. Wash¬ 
ington, D. C., 1886. 

-Physiographic Processes and Features. American Book Co, New 

York, 1896. 

Raymond, Wm. G. A Text-book on Plane Surveying. American Book 
Co., New York, 1896. 

Richards, Col. W. H. Military Topography. Harris & Sons, London, 
1888. 

Schott, Chas. A. Magnetic Declination of the United States. U. S. 
Coast and Geodetic Survey, Appendix 11, Report for 1889. Wash¬ 
ington, D. C. 

-Secular Variation of Magnetic Declination in the United States. 

U. S. Coast and Geodetic Survey, Appendix 7, Report for 1888. 
Washington, D. C. 

-Terrestrial Magnetism. Trans. Am. Soc. C. E., New York, October, 

1893. 

Schweizerische Landesvermessung. Eidg. Topograpischen Bureau, 
Bern, Switz., 1896. 

Smith, Leonard S. Experimental Study of Field Methods which will 
Insure to Stadia Measurements Greatly Increased Accuracy. Bulletin 
University of Wisconsin, Vol. I, No. 5. Madison, Wis., May, 1895. 
Smith, Lieut R. S., U.S.A. A Manual of Topographic Drawing. 

John Wiley & Sons, New York, 1885. 

Specht, George G.; Hardy, A. S ; McMaster, Jno. B.; Walling, Henry F. 

Topographical Surveying. D. Van Nostrand & Co., New York, 1884. 
Teeple, Jared, and Ockerson, J. A. Tables showing Differences in Level in 
Feet, for Horizontal Reduction with Stadia. Engineering Department, 
U. S. Army. Government Printing Office, Washington, D. C., 1879. 
Thiery, E. Des Methodes Topographiques. A. Barbier et F. Panlin, 
Nancy, France, 1902. 

Tittmann, O. H. Precise Leveling, Instruments and Methods. Appen- 





494 


TOPOGRAPHIC DR A WINC 


dices 15 and 16 of the U. S. Coast and Geodetic Survey, Report for 
1879. Washington, D. C. 

U. S. Coast and Geodetic Survey. Topographic Conference, Appendix 
No. 16, Report for 1891. Washington, D. C. 

Van Ornum, J. L. Reduction Formulae for Stadia Leveling. Journal 
Association of Engineering Societies, Vol. XII, No. 8. Chicago, 
August, 1893. 

Van Ornum, J. L. Topography of the Survey of the United States 
Boundary. Trans. Am. Soc. C. E., Vol. XXXIV, No. 4. New 
York, October, 1895. 

Verner, Capt. Willoughby. Rapid FieM Sketching and Reconnaissance. 
W. H. Allen & Co., London, 1889. 

Wainright, D. B. Revision of A Treatise on the Plane-table, by E. Her- 
gesheimer. Appendix 7, Report of U. S. Coast and Geodetic Survey. 
Washington, D. C., 1905. 

Walling, Henry F.; McMaster, Jno. B.; Hardy, A. S.; Specht, George G. 
Topographical Surveying. D. Van Nostrand & Co., New York, 
1884. 

Wellington, A. M. The Economic Theory of the Location of Railways. 
John Wiley & Sons, New York, 1887. 

Wheeler, Capt. Geo. M. Report on U. S. Geographical Surveys. Dept, 
of Engineers, U. S. A. Washington, D. C., 1889* 

Williamson, Lieut.-Col. R. S. Treatise on Use of Barometer. Profes¬ 
sional Papers, Corps of Engineers, U. S. A., No. 8. Washington, 
D. C., 18—. 

Wilson, Herbert M. Topographic Map of the United States. Trans. 
Am. Soc. C. E., Vol. XXXIII, No. 5. New York, 1895. 

-Spirit-leveling. Trans. Am.. Soc. C. E., Vol. XXIV, No. 1. New 

York, January, 1898. 

Winslow, Arthur. Stadia Surveying. D. Van Nostrand & Co., New 
York, 1884. 

Winston, Isaac. Precise Leveling-rods. Appendix 8, U. S. Coast and 
Geodetic Survey, Report for 1895. Washington, D. C. 

Woodward, R. S. Smithsonian Geographical Tables. Smithsonian 
Institution, Washington, D. C., 18—. 



PART V. 

TERRESTRIAL GEODESY. 


CHAPTER XXI. 

, * \ <v - * . ' ... “ f . 

FIELD-WORK OF BASE MEASUREMENT. 

201. Geodesy. —Geodesy has been defined as a system of 
the most exact land measurements, extended in the form of a 
triangulation over a great area, controlled in its relation to 
the meridian by astronomic azimuths computed by formulae 
expressed in the dimensions of the spheroid, and placed in 
its true position on the surface of the earth by astronomic 
latitudes and differences of longitude from an established 
meridian. 

Geodesy in its most general sense may be more briefly 
defined as the solution of problems which are conditioned by 
considerations of the figure and dimensions of the earth. 
Those particular problems which occur in plane and topo¬ 
graphic surveying are solved without regard to the form of 
the earth. (Art. 52.) 

Geodetic operations include, in the order given : 

1. The determination of the exact length, by measure¬ 
ments reduced to mean sea-level, of a line several miles in 
length, which is the base line of the triangulation; 

2. The determination of the latitude and longitude bf one 

495 



49^ FIELD-WORK OF BASE MEASUREMENT. 

end of the base line and of the azimuth of the line by astro¬ 
nomic observations; 

3. The expansion of the base by triangulation executed 
with theodolite; and 

4. The computation of the triangulation, whereby the 
geodetic coordinates of each of the trigonometric points are 
determined. 

To these may be added: 

5. The measurement and computation of geodetic co¬ 
ordinates of controlling points upon a route traverse, adjusted 
or reduced to one or more astronomic positions. 

One of the primary objects of geodetic operations is to fur¬ 
nish data for the exact reference of a topographic map to its 
corresponding position upon the surface of the earth. 

This consists of the measurement of a base line (Art. 202), 
which is an arbitrary distance upon the surface of the earth, 
to which the remaining surveyed positions may be referred in 
standard units, as meters or miles. Also "the determination 
of the astronomic position upon the earth’s surface (Part VI) 
of some initial point on this line, and its azimuth , that it may 
be platted in correct relation upon the map. 

Geodetic operations are also executed for the purpose 
of checking astronomic positions determined by systems 
of primary triangulation or traverse extended from some 
point the geodetic coordinates of which have been already 
determined. 

Astronomic checks on the quality of geodetic triangulation 
by a single determination of astronomic position are far less 
accurate than the positions obtained and computed by trigo¬ 
nometric operations. It is unnecessary, however, to intro¬ 
duce astronomic checks upon primary triangulation at fre¬ 
quent intervals, though these should be sufficiently numerous 
to eliminate station error. The positions determined by 
primary triangulation (Chap. XXV) are not likely to be in 
error by amounts larger than those introduced by astronomic 


BASE MEASUREMENT. 


497 


observation after the triangulation has been extended a dis¬ 
tance of 150 to 250 miles. Therefore a system of primary 
triangulation should be checked by astronomic observation 
at intervals not greater than this. 

Primary traverse is far more liable to errors than is 
primary triangulation, because of the greater number of 
courses sighted and the consequent opportunity for the 
accumulation of error both in angular and in distance meas¬ 
urement. Primary traverse (Chap. XXIII) must therefore 
be more frequently checked by astronomic determination, 
and such checks should not exceed 100 miles apart. 

Whereas primary triangulation and primary traverse may 
be executed with various degrees of accuracy, according to 
the distance to which a system of such control is to be propa¬ 
gated and according to its objects, astronomic determina¬ 
tions should be of the highest order of accuracy. Only the 
most refined instruments and methods known to science for 
use in the field and in permanent observatory work give 
results of sufficiently high quality to fulfill their purposes. 

202. Base Measurement. —The selection of a site for a 
base line is the first step towards the making of a trigono¬ 
metric survey, and on its proper selection depends much of 
the quality of the subsequent work of triangulation. 

1. The site should be reasonably level; 

2. It should afford room for a base from 4 to 8 miles in 
length; 

3. Its ends must be intervisible and so situated as to per¬ 
mit of the expansion of a system of primary triangulation 
which will form the best-conditioned figures. 

The degree of accuracy with which the base measurement 
is to be made depends upon the uses to which the resulting 
triangulation is to be put. 

1. If intended for geodetic purposes, the measurement 
must be made with the greatest attainable precision. 

2. If intended only as a base for the expansion of triangm 


49§ 


FIELD-WORK OF BASE MEASUREMENT. 


lation over a comparatively limited area and for the making 
of a topographic map, this measurement should be made only 
with such care as will attain an accuracy such that its errors 
will not affect the map, although multiplied in the resulting 
triangulation as many times as there are stations. 

3. If intended only as a base for a large-scale topographic 
map of but a few square miles, it will be unnecessary to deter¬ 
mine its geodetic coordinates, as the resulting map may de¬ 
pend upon a plane survey. 

The early method of measuring base lines consisted in the 
employment of wooden rods , varnished and tipped with metal, 
which were supported upon trestles and between the ends of 
which contacts were made with great care. The advantage 
of wooden rods consisted in the fact that their length is but 
slightly affected by temperature, and as they were thoroughly 
varnished they were only slightly affected by moisture. Later 
a more accurate method of base measuring was adopted, con¬ 
sisting in the employment of various forms of compensated rods y 
as the Contact-Slide Apparatus (Art. 210) of the U. S. Coast 
Survey and the Repsold primary base bars of the U. S. Lake 
Survey (Art. 212). More recently the use of steel tapes (Art. 
204) has become popular, as the accuracy attainable with these 
has become better appreciated. The latest approved base bar 
apparatus is the Eimbeck duplex-bars of the U. S. Coast Survey. 
Finally the iced bar (Art. 211) is the highest development 
of base-measuring apparatus adopted by the same Survey. 

203. Accuracy of Base Measurement.—The chief sources 
of error in base measurement, by whatsoever means made, are 
due to— .dr , 

1. Changes of temperature; 

2. Difficulties of making contact; and 

3. Variations of the bars or tape from the standards. 


The refinements of measurement consist especially in— 
a. Standardizing the measuring apparatus or its compari¬ 
son with a standard of length. ~ ' 'T :: 


499 


ACCURACY OF BASE MEASUREMENT . 

r .** V • • .* >.# I .» * . ^ - “ 

b. Determination of temperature or its neutralization by 
the use of compensating bars; and 

c. Means adopted for reducing the number of contacts to 
the fewest possible, and of making these with the greatest 
degree of precision. 

The inherent difficulties of measurement with bars of any 
kind are: 

1. Necessity of measuring short bases because of the 
number of times which the bars must be moved. 

2. Their use is expensive, requiring a considerable number 
of men ; and 

3. The measurement proceeds slowly, often occupying 
from a month to six weeks. 

The advantages of measurements made by a steel tape are: 

1. Great reduction in the number of contacts, as the tapes 
are about three hundred feet long as compared with bars of 
about twelve feet; 

2. Comparatively small cost because of the few persons 

required; > , 

3. Shortness of the time employed, an hour to a mile 
being an ordinary record in actual measurement; and 

4. Errors in trigonometric expansion may be reduced by 
increasing the length of the base from 5 miles, the average 
length of a bar-measured base, to 8 miles, not an uncommon 
length for tape-measured bases. 

Prof. T. C. Mendenhall, in reviewing the qualities of the 
various base apparatus, stated that “The use of an iced bar 
applied to the measurement of considerable distances is un¬ 
questionably the method of highest precision , and its cost is not 
believed to be greater than that of other primary methods in 
use in Europe, but it will not be found necessary to resort to 
it in ordinary practice except for purposes of standardization.” 
He then goes on to state that “the metallic tape is capable 
of giving a result of great accuracy in the hands of experts. 



5 oo 


FIELD-WORK OF BASE MEASUREMENT . 


and that this is evidently the best device for rapid base 
measurement when no great precision is aimed at.” 

It seems that the steel tape is capable of giving a preci¬ 
sion indicated by a probable error of g-g o"UTo o' P art a meas_ 
ured line, while appears to be easily and cheaply 

attainable with long tapes after they are standardized. This 
is amply sufficient for the present purposes of geodesy, and 
the sole obstacle in the way of much higher precision, should 
it be deemed essential, appears to be only the difficulty of 
measuring the temperature of the tape. 

Bases are not measured solely for the accuracy attainable 
within themselves, but to attain the greatest accuracy which, 
when expanded through a scheme of triangulation, will not 
introduce into it errors of appreciable amount. Therefore it 
is scarcely economic to strive at an accuracy which will be 
greatly in excess of that attainable in the succeeding triangu¬ 
lation (Art. 240). Precision of measurement represented by 
probable errors of -g-goV0 o' to To 0V0o' sufficient for all prac¬ 
tical requirements of good primary triangulation not required 
in the solution of geodetic problems. 

204. Base Measurement with Steel Tapes. — Steel 
tapes offer a means of measuring base lines which is superior 
to that obtained by measuring bars because (Art. 203) it com¬ 
bines the advantages of great length and simplicity of 
manipulation, with the precision of the shorter laboratory 
standards, providing only that means be perfected for elim¬ 
inating the errors of temperature and of sag in the tape. Base 
lines can be so conveniently and rapidly measured with long 
steel tapes as to permit of their being made of greater length 
than has been the practice with lines measured by bars, and 
as a result still greater errors may be introduced in tape-meas¬ 
ured bases and yet not affect the ultimate expansion any more 
than will the errors in the latter, because of the greater length 
of the base. Primary base lines have been measured by 
means of long steel tapes within recent years by the U. S. 


STEEL TAPES. 


501 


Geological Survey and the U. S. Coast and Geodetic Survey, 
as well as by the Missouri and the Mississippi River Com¬ 
missions, and in each case with satisfactory results. 

As showing the quality of this work, some measures made 
by Prof. R. S. Woodward of the St. Albans base, previously 
measured with base bars by the U. S. Coast and Geodetic 
Survey, showed a range of several measurements of the whole 
base of 24.1 millimeters or tfoVito °f the whole, the greatest 
divergence from the adopted mean being 13 millimeters or 
S'Tgwtro- The probable error of one measure of a kilometer 
made at this time was ± 1.88 mm. It is believed that 
the probable error of a single tape length did not exceed 
± 0*05 mm., while the probable error of the measurement of 
the whole base was 

205* Steel Tapes. —The tapes used for this work are of 
steel, either 300 feet or 100 meters in length. The meter 
tapes used by the Coast Survey are 101.01 meters in length, 
6.34 millimeters by 0.47 millimeters in cross-section, and weigh 
22.3 grams per meter of length. They are subdivided into 
20-meter spaces by graduations ruled on the surface of the tape, 
and their ends terminate in loops obtained either by turning 
back and annealing the tape on itself or by fastening them 
into brass handles. When not in use the tapes are rolled on 
reels for easy transportation. 

The steel tapes used by the Geological Survey are similar 
to those used by the Coast Survey, excepting in their length, 
which is a little over 300 feet. They are graduated for 300 
feet and are subdivided every 10 feet, the last 5 of which at 
either end is subdivided to feet and tenths. The various in¬ 
strument-makers now carry such tapes in stock, wound on 
hand-reels. All tapes must be standardized before and after 
use by comparison with laboratory standards, and, if possible, 
thereafter frequently in the field by means of an iced bar ap¬ 
paratus. (Art. 2 11.) 

206. Tape-stretchers. —In measuring with steel tapes a 



502 


FIELD-WORK OF BASE MEASUREMENT. 


uniform tension must be given. In order to get a uniform 
tension of 20 to 25 pounds some form of stretcher should be 
used. That used by the U. S. Coast Survey consists of a 



Fig. 152.—Coast Survey Tape-stretcher. 


base of brass or wood, 2 or 3 feet in length by a foot in 
width, upon which is an upright metallic standard, and to this 
is attached by a universal joint an ordinary spring-balance, to 
















TA PE- S TEE TCHERS. 5 O 3 

which the handle of the tape is fastened. (Fig. 152.) The 
upright standard is hinged at its junction with the base, so 
that when the tape is being stretched the tapeman can put 
the proper tension on it by taking hold of the upper end of 
the upright standard and using it as a lever, and by pulling it 
back toward himself he is enabled to use a delicate leverage 
on the balance and attain the proper pull. 

The thermometers used are ordinary glass thermometers, 
around the bubbles of which should be coiled thin annealed 
steel wire, so that by passing that in the air adjacent to the 
tape a temperature corresponding to that of the tape can be 
obtained. Experience with such thermometers shows that 
they closely follow the temperature of the steel tape. For the 
best results two thermometers should be used, each at about 
one-fourth of the distance from the extremities of the tape. 

The stretching device used by the U. S. Geological Sur¬ 
vey is much simpler and more quickly manipulated than that 
of the Coast Survey. It is also more simple than, and, it is 
believed, equally as satisfactory as, that employed by the 
Mississippi River Commission, in which a series of weights 
are employed to give a proper uniform tension. The chief 
object to be attained in tension is steadiness and uniformity 
of tension; the simplest device which will attain this end is 
naturally the best. Two general forms of such devices are 
employed by the U. S. Geological Survey, one for measure¬ 
ment of base lines along railways, where the surface of the 
ties or the roadbed furnishes support for the tape, and the 
device must therefore be of such kind as to permit of the 
ends being brought close to the surface; the other is em¬ 
ployed in measure made over rough ground, where the tape 
may frequently be raised to considerable heights above the 
surface and be supported on pegs. 

The stretcher used by the Geological Survey for measur¬ 
ing on railways/'is illustrated in Fig. 153, and was devised 
by Mr. H. L. Baldwin. It consists of an ordinary spring- 


504 


FIELD-WORK OF BASE MEASUREMENT. 



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Fig. 153.—Tape-stretcher for use on Railroads, 



















































































LA YING OUT THE BASE. 


50 S 


balance attached to the forward end of the tape, where a 
tension of twenty pounds is applied, the rear end of the tape 
being caught over a hook which is held steadily by a long 
screw with a wing-nut, by which the zero of the tape may be 
exactly adjusted over the mark scratched on the zinc plate. 
The spring-balance is held by a wire running over a wheel, 
which latter is worked by a lever and held by ratchets in any 
desired position, so that by turning the wheel a uniform 
strain is placed on the spring-balance, which is held at the 
desired tension by the ratchets. 

The tape-stretcher used by the U. S. Geological Survey 
off railways consists of a board about 5 feet long, to the for¬ 
ward end of which is attached by a 
strong hinge a wooden lever about 5 
feet in length, through the larger por¬ 
tion of the length of which is a slot 
(Fig. 154). Through the slot is a bolt 
with wing-nut, which can be raised or 
lowered to an elevation corresponding 
with the top of the hub over which Fig. 154.— Simple Tape- 

. , • 1 , , STRETCHER. 

measurement is being made, and hung 

from the bolt is the spring-balance, to which the forward 
tapeman gives the proper tension by a direct pull on the 
lever, the weight of the lever and the friction in the hinge 
being such as to make it possible to bring about a uniform 
tension and to hold that tension without difficulty. The 
zero on the rear end of the tape is adjusted over the contact 
mark on the zinc by means of a similar lever with hook-bolt 
and wing-nut, but without the use of spring-balance. 

207. Laying out the Base. —The most laborious opera¬ 
tion in base measurement is its preliminary preparation or the 
“ laying out ” of the base, as it is called, which consists of— 

1. Aligning it with a theodolite; 

2. Careful preliminary measurement for the placing of 
stakes on rough ground ; and 











So6 Field-work of base measurement . 

3. Placing of zinc marking-strips on the stakes and, in 
the case of railway measurement, on the ties. 

Where a base is measured on a railway tangent, no align¬ 
ment is needed beyond a provision for keeping the tape at a 
uniform distance from one rail. In measuring along rail¬ 
ways, a number of boards 5 feet in length, and equal to the 
number of tape-lengths to be laid down, are provided, and 
nailed across the ties at the proper distances. Numbered 
strips of zinc, 6 to 18 inches in length and an inch in width, 
are tacked to blocks of wood nailed on the boards. The 
latter form the support for the tension device, and the con¬ 
tacts are scratched upon the strips of zinc. The thermometers 
by which the temperature is observed are wound with fine 
wire, and at least two are used by which' careful readings are 
made for each tape-length. The base is invariably measured 
at least twice, and the two results are compared by sections 
of at least four tape-lengths. The measurements are prefer¬ 
ably made at night or on very dull and cloudy days, and after 
the line has been once prepared a base of about 5 miles in 
length can be measured in as many hours. 

Base lines measured with steel tapes across country are 
aligned by theodolite, and are laid out by driving large hubs 
of 3 X 6 scantling into the ground, the tops of the same pro¬ 
jecting to such a height as will permit a tape-length to swing 
free of obstructions. These large hubs are placed by careful 
preliminary measurement at exact tape-lengths apart, and be¬ 
tween them, as supports, long stakes are driven at least every 
50 feet. Into the sides of these near their tops are driven, 
horizontally, long nails, which are placed at the same level by 
eye, by sighting from one terminal hub to the next. On 
these nails the tape rests, and on the surface of the terminal 
hubs are tacked strips of zinc on which to make the contact- 
marks. A careful line of spirit-levels must be run over the 
base line, and whether measured on a railroad or on rough 
ground the elevation of the hub or contact-mark of each tape- 


MEASURING THE BASE. S°7 

length must be determined in order to furnish the data for 
reduction, both for slope and to sea-level. 

208. Measuring the Base. —A party for the measure¬ 
ment of a base line along a railway consists of four men : the 
chief of party, who marks the front extremity of the tape and 
has general supervision of the work; a rear chairman, who 
adjusts the rear end of the tape to the contact-marks, and 
reads one thermometer; the head chainman, who adjusts the 
forward end of the tape, applies the requisite tension, and 
reads a second thermometer; and a recorder. In measuring 
over rough ground off railways six men are necessary, namely, 
two tape-stretchers, two markers, two observers of thermom¬ 
eters, one of whom will record. The cooperation of these 
men is obtained by a code of signals, the first of which calls for 
the application of the tension, then the two tape-stretchers 
by signal announce when the proper tension has been applied ; 
then the rear observer adjusts the rear graduation over the 
determining mark on the zinc plate and gives a signal, upon 
hearing which the thermometer-recorder near the middle of 
the tape lifts it a little and lets it fall on its supports, thus 
straightening the tape. Immediately thereafter the front 
observer marks the position of the tape graduation on the zinc 
plate, and at the same time the thermometers are read and 
recorded. By this method a speed can be obtained as great 
as six to eight miles per day. 

209. Compensated Base Bars. —Compensated base ap¬ 
paratus consists of two bars of different metals which have 
different rates of expansion, laid close together, parallel and 
firmly fastened together at the center, from or to which point 
they are free to expand or contract. At a fixed temperature 
they are taken of the same length, so that if they experience 
an equal change in temperature the lines drawn parallel to 
their extremities will remain always at the same constant 
distance apart. The two bars , one of iron and one of brass, 
are each 10 feet long, f inch in thickness, and if inches 


508 field-work of base measurement. 

in width, and are placed i.i inches apart, connected at the 
centers by two transverse steel cylinders not quite in contact. 
At each extremity of the bars is a metal tongue so connected 
by pivots to the bars as to admit of free expansion. These 
tongues are each 6 inches long, and on a silver pivot at one 
end is marked the compensation point. This compound bar 
is placed in a wooden box and is kept from moving lengthwise 
by means of a brass stay firmly fixed to the bottom of the 
box at the center. A long level is fixed to the upper surface 
of the brass bar and is read by means of a glass-covered open¬ 
ing in the top of the box. The tongues carrying the compen¬ 
sation points project beyond the box, but are carefully pro¬ 
tected, and these points lie in the line of measurement. 

In measuring a base six sets of bars are used, and each 
when in use is supported at one-fourth and three-fourths of its 
length by means of strong brass tripods having rollers on their 
upper surface and provided witff a tangent screw for commu¬ 
nicating a longitudinal motion to the bar, and other screws 
for communicating a transverse motion, and an elevating- 
screw for final adjustment of the level. These tripods rest 
on trestles which are at various heights according to the nature 
of the ground. The interval between two adjacent compen¬ 
sating points lying in a line is brought to exactly 6 inches by 
means of a compensation microscope. 

210. Contact-slide Base Apparatus. —The contact- 
slide base-measuring apparatus, made by Saegmuller & Co., 
consists of two measuring-bars, each 4 meters in length and 
supported on trestles. (Fig. 155.) Th <z measurement is made 
by bringing these bars successively in contact, which is effected 
by means of a screw motion and defined by the coincidence 
of lines on the rod and contact-slide. Each bar consists of 
two pieces of wood about 8 x 14 cm. square and a little less 
than 4 meters long, firmly screwed together. Between the 
pieces of wood is a brass frame carrying three rollers, on the 
central one of which rests a steel rod about 8 mm. in diam- 


CO A' TA CT-SLID E BASE APPARATUS. 


5°9 



Fig. 155. —Contact-slide Bask Apparatus, 








































































































































































































5io 


FIELD-WORK OF BASE MEASUREMENT. 


eter. On each side there is a zinc tube 9 mm. diameter. 
The rod and tubes are supported throughout their length on 
similar systems of rollers. The zinc tubes form with the steel 
rod a metallic differential thermometer, and are so arranged 
that one tube is secured to one end of the rod, being free to 
expand in the other direction, the other tube being in a like 
manner fastened to the other end of the rod. The zinc tubes, 
therefore, with any change of temperature, expand or contract 
in opposing directions, and the amount by which the expan¬ 
sion of the zinc exceeds that of the steel is measured by a 
fine scale attached to the rod, while the zinc tube carries a 
corresponding vernier. The cut shows this arrangement, 
which is identical on both ends 'of the bars; a perforation in 
the wood of the bar allows this scale to be read. In addition 
to these metallic thermometers a mercurial thermometer is at¬ 
tached to the bar about midway of its length. 

The rods and tidies thus forming a united whole are mova¬ 
ble lengthwise on the rollers by means of a milled nut working 
in threads cut on the steel rod, which passes through a circu¬ 
lar opening in the brass plate screwed to the wooden bar, and 
against which the nut presses. Two strong spiral springs 
pull the rods back, and the nut is always pressed against the 
plate. One end of the rod is defined by a plain agate securely 
fastened to it; the other end carries the contact-slide, having 

o 

an agate with a horizontal knife-edge. This slide is a short 
tube, fitting over the end of the rod and pushed outward by 
a spiral spring. A slot in the tube shows an index-plate, 
with a ruled line fastened to the rod. 

To align the bars properly a small telescope is placed on 
each bar, and can be adjusted to bring the line of collimation 
over the axis of the rod. The trestle, shown in the upper 
left-hand corner of the illustration, consists of a strong tripod 
stand, carrying a frame with two upright guides for two cross¬ 
slides, which are separated by a movable wedge. These 
cross-slides can be clamped in any position. By moving the 


ICED-BAR APPARATUS . 


511 

wedge, the bar resting between the uprights is either elevated 
or depressed. To obtain smooth movements, friction rollers 
are provided. To move the bars sideways, a coarse screw 
takes hold of a projection on the lower side of the bar, by 
turning which the bar can be moved laterally. 

There are three pairs of trestles, alike in construction, with 
the exception that the upper slide of the trestle intended for 
the forward end of the bar carries a roller on which the bar 
rests, while the other has a fixed semi-cylindrical surface for 
the support of the bar. In making the measurement, the 
bars being 4 meters in length, the stands are set up at a 
distance of 2 meters, each bar being supported at one- 
fourth its length from the ends, as indicated by painted black 
bands. Each bar has a sector with level alidade attached to 
one side, by which its inclination can be read off to single 
minutes. 

The U. S. Coast and Geodetic Survey has recently used 
with much success a new form of bimetallic contact-slide or 
duplex apparatus designed by Mr. Wm. Eimbeck. This con¬ 
sists of two disconnected bars of brass and steel, of precisely 
similar construction, each 5 meters in length. These are 
reversible and are contained in double metallic truss tubes the 
inner of which is reversible on its axis. They are so arranged 
as to indicate the accumulated difference of length of the 
measures of the brass and steel components. 

211. Iced-bar Apparatus. —This apparatus, which is of 
recent invention and is capable of work of the highest pre¬ 
cision, was designed by Prof. R. S. Woodward. It belongs 
to that type in which a single rigid bar is used as the ele¬ 
ment of length along with micrometer microscopes to mark 
its successive positions. Fig. 157 shows the iced bar in 
cross-section. The measuring-bar is carried in a Y-shaped 
trough , where it is kept surrounded with melting ice. The 
trough is mounted on two cars which move on tracks, and 
the microscopes are mounted on wooden posts which are 


512 


FIELD-WORK OF BASE MEASUREMENT. 




S3NVTd 3XV0V 



t 

3 = 



CO 

tu 

X 


CC 

< 

CO 


I— 

a 

Ld 

co 


o 


CO 

D 

< 

Pi 

< 

a. 

a* 

< 

w 

CO 

■< 

CQ 

X 

w 

►J 

a, 

& 

Q 

« 

o 

w 

n 

S 

w 


o 

m 


0 































































































REP SOLD BASE APPARATUS. 


5fo 

ranged out and set firmly in the ground beforehand, the 
microscopes being clamped or detached from the posts easily 
in moving forward as the measuring of the line progresses, 
and the Y trough being likewise rolled forward on cars over 
a temporary track. The apparatus is 5 meters long, the 
microscope posts being set 5 meters apart, and the supports 
for the car-track a like distance. In field-work the micro¬ 
scopes are shielded by umbrellas instead of by temporary 
sheds, as in the illustration. 

The measuring-bar is a rectangular bar of tire-steel 5.02 
meters long, 8 mm. thick, and 32 mm. deep. The upper 
half of the bar is cut away for 
about 2 cm. at either end to re- 3 
ceive the graduated plates of plat¬ 
inum-iridium, which are inserted 
so that their upper surface lies in 
the neutral surfaces of the bar. 

Three lines are ruled on each of 
these plugs, two in the direction 
of and one transverse to the length 
of the bar. The Y trough sup¬ 
ports the bar, keeps it aligned, 
and carries the ice essential to the 
control of the temperature of the 
bar. It is made of two steel 
plates 5.14 m. long, 25.5 cm. 
wide, and 3 mm. thick. They 
are bent to a Y shape, angle of 
60 degrees, and riveted together at the stem. The bar is 
supported at every half-meter of its length by saddles, as 
shown in the illustrations, and these are rigidly attached to 
the sides of the trough by screws, each saddle carrying two 
lateral and one vertical adjusting-screw. 

When the apparatus is in use the Y trough is completely 
filled with pulverized ice, the upper surface of which is 



Y TROUGH 


1 1! 

J 11 

: 1 s 

-7 

1 • > 

i ■ 4 

dj £ 

1 ' 1 

1 j i 

1 ! 

© 


©p 

_* 


Fig. 157 . —Cross-section of 
Iced-bar Apparatus. 





































5i4 


FIELD-WORK OF BASE MEASUREMENT. 


rounded to about the height shown by the curve in the dia¬ 
gram, that is, to the top of the trough. The amount of ice 
required for this purpose is about 8 kg. per meter of the bar’s 
length. By reason of the sloping sides of the trough, the 
ice is kept in close contact with the bar, especially as the 
trundling of the car produces sufficient jarring to overcome 
any tendency of the ice to pack. An essential auxiliary to 
the apparatus is an ice-crusher and a plane for shaving fine ice 
to pack the ends of the bar. The micrometer microscopes 
which define the successive positions of the bar in measuring 
a line are similar to those used in the Repsold base-measuring 
apparatus. (Art. 212.) ' 

212 . Repsold Base Apparatus. —This is an unusual ap¬ 
paratus and has been used in this country in measuring 
primary base lines of the U. S. Lake Survey. The follow¬ 
ing description of it is copied from the final report of that 
organization : 

This consists of a measuring-bar of steel approximately 
4 meters long. (Fig. 158.) Its exact length at any tem¬ 
perature is known. By the side of the steel bar is a similar 
zinc bar. The two are fastened firmly together in the middle. 
Their unequal expansion is observed upon scales at both ends, 
making a metallic thermometer by which the temperature of 
the steel bar becomes known. These two bars are placed 
within a hollow iron cylinder, called the tube-cyUnder , which 
supports them rigidly and protects them from sudden changes 
of temperature. The bars are supported in the cylinder by 
a system of rollers which keeps them straight, parallel, and 
at constant distance from each other. The combination of 
the two bars and the tube-cylinder is called a tube. The 
tube is provided with a sector which indicates the deviation 
of the tube from the horizontal, so that a base can be meas¬ 
ured upon slightly inclined as well as upon level surfaces. A 
telescope is also attached, which points in the same direction 


BASE LINES: COST, SPEED, AND A CCURA C Y. 5 I 5 


as the tube and enables consecutive tube measurements to be 
kept in the same vertical plane. 

In measuring a base the rear end of the tube is placed at 
the beginning of the line and the position of the front end is 
marked. Then the tube is carried forward and the rear end 
is placed at the mark and the front end is marked again, and 
so on, in the same way that a line is measured with a chain 
and pins. In order that the tube may stand firmly it is sup¬ 
ported upon iron stands, one at each end. These stands 
have three legs, which rest upon iron pins driven in the 



ground. To place the tube exactly in the line and at a 
proper height, the tops of the tube-stands are provided with 
movements in three directions, by means of which the tube 
can be moved sidewise, lengthwise, and up and down. For 
convenience there are four tube-stands, so that two can be 
placed in position while the tube is resting on the other two. 

The positions of the ends of the tube are marked with 
microscopes. Thus while the tube does the work of a chain, 
the microscopes do that of the pins. The microscopes are 
mounted upon iron stands, which, like the tube-stands, are 












































516 


FIELD-WORK OF BASE MEASUREMENT . 


supported upon iron pins driven into the ground. The 
microscope-stands are so constructed that the microscope can 
be placed directly over the end of the tube. The microscopes 
are provided with two motions, so that they can be moved a 
short distance along the line or at right angles to it. They 
also have levels attached, so that they can be made vertical. 
For convenience there are four microscopes, so that two can 
be placed in position while two are standing over the ends of 
the tube. 

To measure a tube-length , the rear end of the tube is placed 
under the microscope which marks the position of the front 
end of the preceding tube-length. The tube is then brought 
into the line by means of its telescope. Its inclination is 
found by reading' its sector, and the temperature of the steel 
bar is found by microscope readings on the scales at front 
and rear ends. From the temperature the length of the steel 
bar is found at the instant the measurement is made. From 
its inclination the horizontal projection of this length is 
found, and thus the actual advance becomes known. 

213. Base Lines: Cost, Speed, and Accuracy. —Base 
lines of the highest attainable accuracy of measurement, as 
those measured by the U. S. Coast and Geodetic Survey, 
cost from twenty-five hundred to thirteen thousand dollars, 
according to the methods employed and the precision aimed 
at. The speed of measurement by the U. S. Coast Survey, 
using base bars, is from two to six months, including prepa¬ 
ration and actual measurement. The probable error of the 
result attained is from to 

Base lines as measured by the U. S. Geological Survey 
with sufficient accuracy for the expansion of primary triangu¬ 
lation which is to be developed to distances of 200 to 400 
miles, cost from one hundred to two hundred dollars per 
base. This work is executed with steel tapes and requires 
from seven to ten days for preparation and measurement of 
the base. The accuracy attained has an average probable 
error of xtfTnnnr* 




CHAPTER XXII. 

COMPUTATION of base measurement. 

214. Reduction of Base Measurement. —After the meas¬ 
urement of the base line has been completed in the field, the 
results of the measurement have to be reduced for various 
corrections, among which are: 

1. Comparison with standard of measure; 

2. Corrections for inclination and sag of tape if such is 
used ; 

3. Correction for temperature; 

4. Reduction to sea-level. 

As an example of the method of making such corrections 
and of keeping the records of base measurement, the follow¬ 
ing has been selected from the measurement of the Spearville 
base in Kansas by the U. S. Geological Survey. 

215. Reduction to Standard. —The first correction to be 
applied is that of reducing the tape-line or the base-bars to a 
standard. The data for this reduction is best obtained by a 
comparison with the international standards of the U. S. Coast 
and Geodetic Survey, that at their office in Washington, or the 
standard kilometer measured at the Holton base in Indiana. 
Or, if these are not accessible, comparisons may be made 
with standards in the possession of the U. S. Mississippi and 
Missouri River Commissions and of one or two of the more 
reliable instrument-makers of the country. 

The reduction to standard may be positive if the tape is 
longer than the standard, or negative if shorter, and this reduc¬ 
tion is proportioned to the entire length of the line, and is 
generally made by multiplying the length of the tape or bar 
as obtained from comparison with the standard into the 

517 


5 18 COMP U TA T10N OF BA SE ME A S U RE MEN T. 

number of times which the same is applied on various sections 
or the whole of the base. 

An example of this reduction is given in Art. 217, which 
contains a record of the measurement of the Spearville base. 

216. Correction for Temperature. —As the length of a 
steel tape or a metal bar varies with the temperature, one of 
the most uncertain elements in the measurement of a base by 
means of a steel tape is its length as compared with the stand¬ 
ard because of variations due to expansion and contraction 
from changes in temperature. As already shown, the most 
accurate mode of measurement obtainable is that in which the 

t 

temperature is fixed, as in the case of the iced-bar apparatus 
(Art. 211). In any other form of base-measuring apparatus 
every effort must be made to obtain the greatest uniformity 
of temperature, and in using a tape corrections must be made 
for every tape-length, as derived from readings of one or 
more thermometers applied to the tape in the course of the 
measurement. 

Steel expands .0000063596 of its length for each degree 
Fahrenheit of temperature. This fraction, multiplied by the 
average number of degrees of temperature above or below 62 
degrees at the time of the measurement, gives the proportion 
by which the base is to be diminished or extended on account 
of temperature changes. This correction is applied usually 
by obtaining with great care the mean of all thermometric 
readings taken at uniform intervals of distance during the 
measurement. An example of the record of temperature and 
of reduction for temperature is given in the last two columns 
of the table in the following article. 

217. Record of Base Measurement. —The following is a 
sample page from the note-book containing record of meas¬ 
urement of the Spearville Base as made by Mr. H. L. Baldwin 
of the U. S. Geological Survey in 1889. This base was 
measured along a railroad and therefore has no correction for 
sag as the tape rested on the ties. It was measured in a 


CORRECTION FOR INCLINATION OF BASE. 519 

number of sections, and the following is that of two measure¬ 
ments of the first section. In the last column is shown the 
method of making corrections to the tape to reduce to stand¬ 
ard (Art. 215), and in the next to the last column the method 
of making temperature correction (Art. 216). 

RECORD OF BASE MEASUREMENT AND REDUCTION. 


(First measurement, section 1, October 16, 18S9.) 
H. L. Baldwin, Topographer. 


No. of 
Tape. 

Time, 

A.M. 

Ten¬ 

sion. 

Thermom¬ 

eters. 

Temperature Cor¬ 
rection. 

Total Length of Section. 

A. 

B. 


h. m. 

Pounds 

O 

O 



I 

10 13 

19-75 

5°-5 

50.0 



2 

20 

20.00 

5 °- 5 

50.0 



3 

26 

20.00 

50-5 

50.0 



4 

31 

20.25 

5 ° 5 

5c.0 

Mean temp. = 50°.51 


5 

37 

20.00 

5°-7 

50.5 


1 tape-length. — 300.0617 

6 

42 

20.125 

5 l -5 

50.6 

62° — 5o°.5i = ii°.49 


7 

47 

20 25 

51.0 

50.8 


10 X 300' 0617. = 3,000'.617 

8 

51 

20.00 

50.8 

50.2 

— ii°. 49 X 3000'. 

Temperature coir.. — .207 

9 

55 

20.125 

50.8 

50.0 

,X .000006 


IO 

58 

20.00 

5 o -7 

50.5 

= — '.207 

Result first meas... = 3,000.410 


(Second measurement, October 17, 1889.) 


No. of 
Tape. 

Time, 

P.M. 

Ten¬ 

sion. 

Thermom¬ 

eters. 

Temperature Cor¬ 
rection. 

Total Length of Section. 

A. 

B. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

JO 

h. m. 
12 13 

21 

25 

29 

33 

3 6 

38 

41 

45 

50 

Pounds 

20.00 

20.25 

20.00 

19-75 

20.00 

20.00 

20.00 

20.12 

19 75 
20.13 

0 

5 2 - 3 

53 - 3 
53-8 
55 -o 
55 -o 

53 - 8 

54 - ° 

54 - 5 

55 - 1 
54-5 

0 

52 - 4 

52.9 

54.0 

54-8 

53 - 2 

54 - o 
54-0 

54 -o 

54-4 

54 -i 

Mean = 53°.96 

62° — 53°.96 = 8 °.o 4 

— 8°.04 X 3000'. 

X .000006 
= - / - I 45 

Tape set back from sta. 0 

.85 inch. = .071 foot. 

10 X 300'.0617.= 3,000'.617 

Set back. — .071 

Temperature corr.. — .145 

Result sec. meas.. = 3,000.401 


218. Correction for Inclination of Base—The data for 
this correction are obtained by running a careful line of spirit- 
levels over the base line (Chap. XV). In the course of this 
leveling, elevations are obtained for every plug upon which 
the tape rests. The result of this leveling is to give a profile 
showing rise or fall in feet and fractions thereof between the 
points of change in inclination of the tape-line. From this 



















































520 COMPUTATION OF BASE MEASUREMENT. 

and measured distances between these points the angle of 
inclination is computed by the formula 

sin 6 — .(4°) 

in which D — length of tape or measured base; 

h — difference in height of the two ends of the tape 
or measured base, both in feet; and 
6 — angle of slope expressed in minutes. 

The correction in feet to the distance is that computed 
by the equation 

Cor. = D^± 6 ' .(41) 

2 

sin 2 i 1 

Since, however, -= 0.00000004231, we have 

2 

Cor. = 0.0000000423 1 6 *D .(42) 

As the logarithm of the constant is equal to 2.6264222, 
the above may be expressed in logarithms, thus: 

Cor. = log 2.6264222 -f- 2 log 6 -j- log D. . (43) 

An example of the record and mode of making correc¬ 
tions for inclination is given in the following, taken from 
Spearville Base measurement: 


S. S. Gannett, Computer. 


Approxi¬ 

mate 

Distance. 

Differ¬ 
ence of 
Elevation 

Angle 9 

log 0 

2 log 0 


log 
sin 2 i' 


log dist. 

log 

correc¬ 

tion. 

Correc¬ 

tion. 

2 

feet. 

feet. 

/ // 







0,200 

0.8 

13 34 

1.1326 

2.2652 

2.6264 

2.30 IQ 

7.1926 

• 0015 

4,200 

4.2 

2 22 

0.3674 

0.7348 

1 



4.6232 

6.9844 

.OOIO 

4,000 

12.0 

10 08 

I.0052 

2.0104 




3.6021 

8.2389 

■0173 

1,000 

1.0 

3 23 

0.5250 

1.0501 




3.0000 

6.6765 

.0005 

2,000 

3 -° 

5 04 

0.7024 

1.4049 




3.3010 

7.3323 

.0021 

4.020 

22.0 

12 23 

1.0917 

2.1834 




3.6232 

8.4330 

.O27I 

2,800 

7.0 

8 27 

0.9263 

1.8527 




3-4472 

7.9263 

.0084 

1,000 

0.0. 

O OO 

0.0000 

0.0000 


Con- 


3.ocoo 

O OOOO 

.OOOO 

1,000 

1.0 

3 2 3 

0.5250 

I.0500 


*" stant/ 


3.0000 

6.6764 

.0005 

4,200 

20.0 

11 16 

1.0504 

2.1008 




3.6232 

8 • 3504 

.6224 

3,800 

6.0 

5 20 

0.7267 

1-4535 




3-5798 

7-6597 

.0046 

2,000 

4.0 

6 45 

0.8293 

1.6586 




3.3010 

7.5860 

.0038 

5 i 4 °° 

3 1 • 4 

19 39 

1-2934 

2.5867 




3-7324 

8-9455 

.0882 

2,000 

2.6 

4 24 

0.6437 

1.2874 




3.3010 

7.2148 

.0016 

0,13s 

0.05 

I l8 

0.1072 

0.2144 




2.1303 

4.9712 

.OOOO 









.1790 










































CORRECTION FOR SAG. 


521 


An approximate formula for reducing distances measured 
on sloping ground to horizontal is expressed by the rule: 
Divide the square of the difference of level by twice the 
measured distance, subtract the quotient thus found from 
the measured distance, the remainder equals the distance 
required ; or, 

Ji 2 

d — D ^ jy . (44) 

in which d = horizontal or reduced length. This formula 
may be used in reducing the various inclined measures made 
over rough ground in primary traverse (Art. 227). 

Example: Let 50 = length in feet of distance measured 
on slope, 3 = difference in height in feet between two ends of 
measured line, then 

3* = 9; 9 -f- (50 X 2) — .09 (exact formula (41) gives .0908). 

50 — .09 = 49.91 — d — horizontal distance required. 

219. Correction for Sag.—When the base measurement is 
made with steel tape across country, and is, accordingly, not 
supported in every part of its length as on a railway, there will 
occur some change in its length due to sag. As explained in 
Art. 207, where measurement is made off a line of railroad, the 
tape should be rested on supports placed not less than 50 feet 
apart. With supports placed even this short distance apart, 
however, a change of length will occur between them, while 
even greater changes will occur should one or more supports 
be omitted, as in crossing a road, ravine, etc. As tapes are 
standardized by laying them on a flat standard, it is essential 
to determine the amount of shortening from the above causes. 
The following formulas apply: 

Let w — weight per unit length of tape; 
t — tension applied; 
w 




522 


COMP U TA TION OF BA SE ME A S UR E MEN T. 


n — number of sections in which tape is divided by 
supports; 

/ = length of any section ; 

L = normal length of tape or right-line distance 
between n marks when under tension = nl, 
approximately; 

P- 1= reciprocal of product of modulus of elasticity of 
tape by area of its cross-section. 

If a tape be divided by equidistant supports , the difference 
in distance between the end graduations, due to sag, or the 
correction for sag = dL, becomes 

dL = -fca\n x l' — nj*). 


If one or more supports are omitted, then the omission of 
tn consecutive supports shortens the tape by 

-J- i )(w 2 ^ 0*1 ; 


where / is the length of the section when no supports are 
omitted. 

Example: Let n = 6; l — 50 feet; w — .0145 = weight 
in pounds per foot found by dividing whole weight of tape by 
whole length; and t = 20 pounds;—then 


24 


V 6 X 5 ° | 

^0.0145 x 50') 

' ~ 24 

^ 20 / 


which is the length of sag or shortening of each tape-length. 
This correction is always negative. 

If in a certain measure of a base there were 86 full tape- 
lengths, the total correction for sag would be 


.Cor. — 86 X .0162 — 1.393 feet, 

which is quite an appreciable quantity. 

220. Reduction of Base to Sea-level.—The base is 
always measured on a circle parallel to the mean sea-surface 
and raised above it at an elevation the amount of which must 
be known, at least approximately. This circle with radii 
drawn therefrom to the center of the earth forms a triangle 





REDUCTION OF BASE TO SEA-LEVEL. 


523 


approximately similar to that formed by the radii of the earth 
with the sea-surface. The length of the base at sea-level is 
therefore derived with a sufficient approximation to correct¬ 
ness by the proportion 

D H 

r : H :: D : d, or Cor. =-, . . . (45) 


in which r — the radius of the earth ; 

H — the height o: base line above mean sea-level; 

D — the measured length of the base line; 
d — the correction to reduce this measured length to 
length at mean sea-level. 

An example of the form of such reduction is the follow¬ 
ing, taken from the Spearville Base: 


REDUCTION TO SEA-LEVEL. 


Correction. . . . 

log D (meters) 
log H (meters) 
Co log r . 


DH 

r 

- 4.05956 
= 2.87599 
= 3.19660 


log 1.356 meters..= o. 13215 

log. meters to feet.= 0.51599 


4.448 feet (always subtractive) ... .0.64814 

221. Summary of Measures of Sections—Corrections 
for temperature and standard having been made to each of 
the sections of the measured base (Arts. 215 and 216), the 
mean of the several measures of each section must be obtained 
and the total length of the base will then be obtained by sum¬ 
mation of the reduced lengths of sections. The table on page 
524 is an example of the record of such summary of sections. 

222 . Corrected Length of Base.—The foregoing correc¬ 
tions and summations having been made, the correct length of 
the base may now be obtained by applying the corrections 
for inclination and reduction to sea-level both of which are 












524 COMPUTATION OP BASE MEASUREMENT. 

SPEARVILLE BASE: SUMMARY BY SECTIONS. 
(Corrected for Temperature.) 


S. S. Gannett, Computer. 


Stations. 

First 

Measure. 

Second 

Measure. 

- 1 

Difference. 
First — Second . 



feet. 

feet. 

feet. 

I 

to IO 

3,000.410 

3,000.401 

4 - .009 

IO 

20 

.418 

•393 

+ •025 

20 

30 

.431 

•431 

-(- .OOO 

3 ° 

40 

.426 

.446 

— .020 

40 

50 

•437 

.478 

— .041 

50 

60 

.417 

•455 

— .03S 

60 

70 

•369 

•392 

— .023 

70 

80 

.366 

.356 

4 -.010 

80 

90 

• 955 

•938 

+ .017 

90 

IOO 

.676 

.667 

4.009 

100 

no 

3,000.89'9 

3,000.898 

4 - .001 

no 

119 

2.700.581 

2,700.571 

4~ 010 

119 

126 

2,100.244 

2,100 23-| 

-|- .010 


37,806.629 

37,806.660 

— .031 


always negative (Arts. 218 and 220). This is done in the 
following manner: 

MEAN OF TWO MEASUREMENTS. 


Correction for temperature and standard. 37,806.645 ft. 

“ “ inclination. 0.179 “ 

“ “ reduction to sea-level. 4.448 “ 

Final corrected length of measured base. 37,802.018 ft. 


223. Transfer of Ends of Base to Triangulation Sig¬ 
nals. —In the foregoing article the corrected length of the 
base, 37,802.018 feet, would be the final length of the base 
as determined under ordinary circumstances—that is, when 
the ends of the base line are also the astronomic pier and tri¬ 
angulation station from which the expansion of the base is 
made. Occasionally, and especially where a base is measured 
on a railroad, it is impossible to erect the astronomic pier or 
the trianuglation stations (Art. 243) over the extremities of 
the base, and it then becomes necessary to transfer the meas¬ 
ured length of the base to the triangulation signals and pier, 
which are erected as near as possible to each end. 

In the case of the Spearville Base, the astronomic pier 

























TRANSFER OF EASE TO TRIANGULATION SIGNALS. S 2 S 


was first erected at the northeast extremity of the base and a 
triangulation signal was erected near the railroad track at the 
point selected for the southwestern extremity of the base, 
and triangulation was started by reconnaissance and erection 
of signals prior to the measurement of the base line. Accord¬ 
ingly, after the base had been measured its length had to be 
transferred to the triangulation signals. The following is an 
example of the elements of this reduction. 

The end 0 (Fig. 159) was not selected at exactly right 


Fig. 159.— Transfer of Measured Base ( 00 ') to Marked Base (ABJ. 

angles to the pier and station, but at a distance a little be¬ 
yond the extremity of the pier so that the angle between the 
base and the pier was less than 90°. These angles were 
carefully measured at the extremities of the base O and O', 
also the distance from O to the pier at A , and the distance 
a to O. Solution of the right-angled triangle AaO gave 
aO = D — 2.864 feet. At the southwest extremity of the 
base marks were left at the 125th and 126th tape-lengths, and 
the angles read at these points between the measured base line 
and signal B, also the angles at the signal B to those marks, 
the distance between them also being noted as an exact tape- 
length. These data gave the elements necessary for the solu¬ 
tion of the right-angled triangles into which the main triangle 
was divided by the projection of B at right angles to the base: 
line at the point b, and the amount determined by which, the: 
measured base was to be reduced was b -j- 126 = 168.235 feet.. 

The following is the mode of applying this correction to 
the corrected measured base length to get the secondary base 
or the distance between the triangulation signals : 


Corrected length of measured base. 37,802.018 ft.. 

Reduction from southwestern base to triangulation signal... 168.235 “ 

Reduction from northeastern base to triangu'lation station.. 2.864 “ 


Corrected length of triangulation base. 37,630.919 ft. 







$26 COMPUTATION OF BASE MEASUREMENT. 


224. Other Corrections to Base Measurements.—In 

addition to the simple corrections above given, which are 
always made to the measurement of base lines, it is some¬ 
times desirable to determine the modulus of elasticity of the 
metal in order to make corrections for the pull in stretching 
the tape. This correction is, however, often of doubtful 
application, because the exact amount of pull at any time 
may be carelessly noted. It is far better and quite as simple 
to eliminate such corrections by giving a uniform pull at all 
times, thus doing away with the correction for modulus of 
elasticity. Another correction is for metallic thermometers ; 
but as glass thermometers can be purchased without difficulty 
and almost anywhere, it seems unnecessary to make provision 
for such correction. 

225. To Reduce Broken Base to Straight Line —Occa¬ 
sionally, because of some obstacle to the straight alignment 
of the base or in order that either extremity may terminate 
in the most desirable position for the expansion of triangula¬ 
tion, it becomes necessary to introduce one or more angles in 
a base measurement. This, however, should never be done 
unless absolutely unavoidable, and then such angles should 
never deviate greatly from 180 0 . The correction for such a 
broken base may be expressed as follows: 

If the measured base be in two lengths, A and B and it 
being necessary to find the third side of the triangle which 
they form, the latter being the straight line L\ then, #, being 
the difference between the angle and 180 0 , we have 


L — A B — 0.00000004231 


ABO " 2 
A + B' 



This formula cannot be employed, however, where 0 is greater 
than 5 0 , in which case the unknown side will have to be com¬ 
puted by the ordinary sine formula for the solution of tri¬ 
angles. (Chap. XXVII.) 


CHAPTER XXIII. 


FIELD-WORK OF PRIMARY TRAVERSE. 

226. Traverse for Primary Control. —It is frequently 
inexpedient, because of the relative expense, to procure 
primary control for topographic mapping by means of tri¬ 
angulation. Sometimes, especially in heavily forested and 
level country, it is practically impossible to execute primary 
triangulation within any reasonable limits of time or cost. 
The means adopted for securing sufficient primary control 
under such conditions is by the running of traverse lines of 
a high degree of accuracy. 

Primary traverse does not differ from secondary traverse, 
such as is executed for the determination of topographic de¬ 
tails (Art. 87) in the general methods of its execution. It does 
differ therefrom materially in the quality of the instruments 
employed and the elaborateness of detail with which the field 
and office computations are conducted. It may, therefore, 
be likened rather to the measurement of a series of long and 
broken but connected base lines measured in a manner rather 
similar to that explained in Art. 208, but with less care. 

As primary traverse is executed for control of topo¬ 
graphic mapping, it furnishes the initial coordinates by which 
the topographic map is fixed in astronomic position. One or 
more points of the primary traverse must therefore have 
their geodetic coordinates determined by astronomic observa¬ 
tion (Part VI) or by connection with a system of primary 
triangulation (Chap. XXV). Primary traverse is materially 
inferior in quality as control to primary triangulation (Art. 

527 


528 FIELD-WORK OFPRIMARY IRA VERSE. 

201). In order that errors occuring in its execution may be 
reduced, it is desirable that the two extremities, and, if very 
long, a middle point on the primary traverse line, be checked 
either by closing the'traverse back on itself or on some other 
adjusted primary traverse. 

The best and in fact the only satisfactory means of dis¬ 
tributing by adjustment the errors inherent in the primary 
traverse is to connect at least two of its more remote points 
with primary triangulation stations or astronomic positions. 
(Chap. XXV and Part VI.) 

227. Errors in Primary Traverse. —The errors inherent 
in primary traverse are of three general classes: 

1. Those due to measurement of deflection angles or azi¬ 
muth errors; 

2. Those due to linear measurement or errors of distance; 

3. Instrumental errors. 

Probably the most serious errors introduced in primary 
traverse are those due to the measurement of the deflection 
angles or the azimuth errors. These are of several kinds and 
are affected— 

1. By the quality and graduation of the instrument; 

2. By the shortness of the sights; 

3. By the relative dimensions and plumbing of the flag; 

4. By the care exercised in centering the instrument over 
stations. 

The first is to be provided against only by use of such an 
instrument as is best suited to the work to be done and by 
keeping it in perfect adjustment. The second is the most 
important source of error and is not to be confused with the 
irregularity and sinuosity of the traverse run. This may be 
ever so winding, yet, if the sights are sufficiently long and the 
angles not great, the errors by such an irregularity will not be 
of serious moment. These errors are affected by the third and 
fourth factors, and in sinuous traverses of short sights the errors 


ERRORS IN PRIMARY 'ERA VERSE. 529 

in bisecting a large signal or any centering over a station be¬ 
come matters of considerable moment. 

Where such ordinary care is exercised as is indicated in 
Articles 228 and 229, and in the instructions Art. 231, the 
errors of measurement will be relatively small. Likewise, the 
errors of instrument should be small providing the ordinary 
precautions designated for care and adjustment of instruments 
are exercised. 

In the measurement of distance the most important source 
of error is: 

1. Failure to keep the tape horizontal; 

2. Carelessness in plumbing down to center points where 
there is much inclination and short tape-lengths are used ; 

3. Failure to apply a uniform tension ; and 

4. Errors in count or record of number of tape-lengths. 

Where the traverse is over a good line of railroad having 

easy grades and long tangents the best results may be ex¬ 
pected. In such case it is unnecessary to keep the tape 
horizontal by lifting it above the ground, it being sufficient to 
rest it upon the ties. The error of slope on a good railway 
grade is less than that of sag when the tape is held horizon¬ 
tally. Under such circumstances the chief source of error 
is likely to be in the measurement rather than in the azimuth. 
On the other hand, where the traverse is run over rough 
ground the inaccuracies introduced are greatest in amount. 
Then, as in running railroads having short tangents and con¬ 
sequently short sights, a considerable source of error is in the 
azimuths, and even a greater source of error arises from the 
necessity of taking short tape-lengths on sloping ground and 
plumbing down to center marks. It is evident, therefore, 
that not only is greater precision obtained in measuring over 
good lines of railway, but also the speed is materially in¬ 
creased and the cost reduced proportionately. 

228. Instruments used in Primary Traverse —The in¬ 
strument used for measuring azimuths should be a transit of 


530 


FIELD-WORK OF PRIMARY TRAVERSE. 


high grade, having a six- to eight-inch circle and reading to 
20 or 30 seconds. Such an instrument should have a hollow 
telescope axis and be provided with a lamp and other attach¬ 
ments for night-work. As an important source of error in 
such work is in the azimuth, this should be checked nightly, if 
weather permits, by observations on a circumpolar star (Art. 
312) at or near elongation. When the line of traverse is 
crooked such observations should never be at intervals greater 
than ten to fifteen miles. When the route traversed has 
long tangents, distances between check azimuths may be 
increased. 

There are two methods of measuring the horizontal angles : 

1. By transiting the telescope and reading forward deflec¬ 
tion angles as with an engineer’s transit (Art. 87); 

2. By reading full circle or deflection angles from back¬ 
sight to foresight. 

The former is preferable when the instrument is kept in 
good adjustment, as it is more rapid and more accurate. The 
process consists in sighting on rear flag, transiting telescope, 
and revolving on upper circle until fore flag is bisected by the 
cross-hairs. The angle read is the deflection from the last 
sight prolonged to the new sight (Fig. 67). Then the upper 
circle is again revolved through nearly 180° until the rear flag 
is again bisected. Once more the telescope is transited and 
the fore flag bisected. The result is two separate records and 
two measures of the angle, one a single measure, and the 
other double. Moreover, one pointing is with telescope 
direct, and the other with it reversed. 

The second method of measuringthe horizontal deflections 
is to point the telescope at the rear flag and read both ver¬ 
niers as before. Then with lower motion clamped, the instru¬ 
ment is revolved horizontally on the upper plates without 
transiting, and pointed at the fore flag, and both verniers are 
again read. The difference between the two readings is the 
deflection or arc through which the telescope has been re- 


METHOD OF RUNNING PRIMARY TRA TERSE. 531 

volved. By repeating this operation at least two measures 
are made, one with telescope direct, and the other with it 
reversed and should be on different parts of the circle. (Art. 
252.) 

Distances in primary traversing should be measured with 
a three-hundred-foot steel tape of kind similar to those em¬ 
ployed in measuring base lines (Art. 205). The tape should 
be tested by a standard and be corrected for average tem¬ 
perature somewhat as in measuring base lines (Art. 216). 
Every effort should be made to use only even tape-lengths. 
As the tapemen, however, approach the instrument point a 
tape-length must necessarily be broken, and care must be 
exercised in the precautions employed to measure the frac¬ 
tional tape. A good way of doing this is by having a three- 
hundred-foot tape divided by clear markings into hundred- 
foot lengths and then to use a standardized one-hundred-foot 
steel tape for measuring the fraction less than one hundred 
feet. 

229. Method of Running Primary Traverse. —The party 
organization for running a primary traverse should consist of 
five or six persons; namely, the chief as transitman, one re¬ 
corder, two tapemen, and one or two flagmen. The transit- 
man directs the movements of the other members of the party 
and determines directions by reading angles on the transit 
instrument. He also reads the compass-needle as a check on 
the azimuth computation. The recorder keeps a record of 
the angles called off by the transitman, works up the mean 
pointing as the work advances, notes by observation of the 
angles recorded whether any gross error has been made in 
reading of the transit vernier, and calls the attention of the 
transitman to such errors if any exist. He or the chief of 
party also checks the measurement of distance by the tape- 
men by counting rail-lengths or by pacing, recording the same 
in the first or station column as shown in the example (Art. 
230), thus checking the liability of making gross errors. 


53 2 FIELD-WORK OF PRIMARY TRAVERSE. 

About once an hour he also reads a thermometer held beside 
the tape at an instrument station. 

The tapemen measure the distance with the steel tape, 
which is stretched by a twenty-pound tension on the front end 
by the fore tapeman with a spring-balance. Temperature is 
also read and recorded by one of the tapemen, and both tape- 
men keep a record of the number of tape-lengths between 
stations. These distances are worked up daily into the notes 
kept by the recorder. The rear flagman gives backsight for 
the transitman, who aligns one of the tapemen as a fore flag¬ 
man. Or a fore flagman may be employed, when the speed 
will be increased somewhat. 

The initial and terminal points of the primary traverse 
must be well indicated by permanent marks , as should numer¬ 
ous intermediate points on the line of the traverses, especially 
at such places as may be used as tie points for other primary 
or secondary control (Art. 248). All road crossings, stream 
crossings, railway stations, and other permanent objects should 
be indicated in the note-books, that they may furnish check 
points for the control of the topographic or secondary traverse 
(Arts. 14 and 16), and connection with the leveling (Chap. 
XV). 

230. Record and Reduction of Primary Traverse. —The 

following is an example of record and reduction of a portion 
of a primary traverse run near Traskwood, Arkansas, by Mr. 
George T. Hawkins of the U. S. Geological Survey. In the 
first column are given the distances between stations in feet, 
checked by counting rail-lengths; in the following three 
columns are given the readings of the angles recorded by the 
separate verniers and their mean; in next to the last col¬ 
umn is recorded the deflection angle; and in the last column 
are given the computed and corrected azimuths. 

The azimuth recorded in this column in plain type is that 
carried forward by computation from the last station to the 
station occupied, being the algebraic addition to the former 


INSTRUCTIONS FOR PRIMARY TRAVERSE. 


533 


of the deflection angle at the latter. Underneath, in itali¬ 
cized figures, is given the reduced or corrected azimuth which 
is to be used in the further computations. This is obtained 
oy distributing the error found between the last and the next 
observed astronomic azimuth. (Chap. XXXIII.) 


Distance. 

Ver. A 

Ver. B. 

Mean. 

Angle. 

Azimuth. 

Sta. 107 

O / ft 

96 09 OO 
96 09 OO 

Oft/ 

276 09 OO 
276 09 OO 

O t ft 

96 09 OO 
96 09 OO 

0 / // 

O OO OO 

0 OO OO 

O t ft 

44 30 17 

(60 rails) 

1800 feet 

96 09 OO 

276 09 OO 

96 09 OO 
96 09 OO 

0 

44 30 17 

1/1/ 30 10 

Sta. 108 

96 09 OO 
96 09 OO 

276 09 OO 
276 09 OO 

96 09 OO 
96 09 OO 

O OO OO 

O OO OO 


(100 rails) 
3000 feet 



96 09 OO 

0 

44 30 17 

1/1/ 30 01/. 


Brought forward from Sta. 108 to Sta. rj2. 34 43 15 


Sta. 132 

76 27 OO 

256 26 30 

76 26 45 




84 05 OO 

264 04 30 

84 04 45 

7 38 00 





84 04 45 

7 38 30 

42 21 30 

(40 rails) 

1200 feet 

91 43 30 

271 43 OO 

9 i 43 15 

+ 7 38 15 

1/2 18 1/5 

Sta. 133 

9 i 43 30 

271 43 00 

91 43 15 




92 20 OO 

272 19 30 

92 19 45 

0 36 30 





92 19 45 

0 36 45 

42 58 07 


92 56 30 

272 56 30 

92 56 30 

+ 0 36 37 

1/2 55 15 . 


Station 133 4 ~ 130 feet is in Observed azimuth at Traskwood 
front of middle window in between stations 133 and 134. 

Traskwood Depot. 


231. Instructions for Primary Traverse. — The details 
in running primary traverse are best explained in the follow¬ 
ing instructions, which govern the execution of such work by 
the U. S. Geological Survey: 

i. The instruments to be used are a 20" or 30" transit; 
one 300-foot steel tape graduated to feet for five feet at either 
end; one spring-balance; one ioo-foot steel tape; two ther- 































534 


FIELD-WORK OF PRIMARY TRAVERSE. 


mometers ; four hand-recorders; two flagpoles; and one good 
watch. 

2. The party should consist of: One chief, as transit- 
man; one recorder; two tapemen, either of whom may act 
as front or rear flagman ; and one flagman. 

3. At each station the transitman should proceed as fol¬ 
lows: Set telescope on rear flag, read both verniers, transit 
telescope, set on front flag, and read both verniers. Shift the 
circle and remeasure the same angle with telescope reversed. 
If the two angles thus measured differ more than 60", repeat 
the operation. 

4. Along a railroad the operation of measuring is to be 
conducted as follows: The front tapeman puts a 20-lb. ten¬ 
sion on the front end of the 300-foot tape with a spring-bal¬ 
ance. He makes a chalk-mark on the rail, or places a tack or 
nail on a tie, stake, or measuring-board, under the 300-foot 
mark for full tape-lengths, and under the fractional graduation 
at stations. The distance which he records is checked by the 
transitman and at least one other member of the party. The 
tack or nail is left, surrounded by conspicuous chalk-marks, 
and the same process is continued. 

5. The rails should be counted by two others of the party, 
who also check the number of tape-lengths at the first oppor¬ 
tunity. Each station should be marked by a small-headed 
tack or pricking-needle through a piece of white paper or 
cloth, its number being chalked on the rail near where it falls. 
The distance between stations should be limited to the visi¬ 
bility of the flagpoles. Rails or center of track must not be 
used as alignment sights. 

6. Along highways or open country the tape should be 
kept level. On steep slopes a plumb-bob must be used, either 
to bring the tape vertically over an established point or to es¬ 
tablish a new one, as the case may be. Tape-lengths are 
marked on the measuring-board, tie, or stake with the prick¬ 
ing-needle. Where slopes are so steep as to render the level- 


INSTRUCTIONS FOR PRIMARY TRAVERSE 


535 


ing of the 300-foot tape impracticable a shorter tape must be 
used. 

7. The chief and two other members of the party must 
keep an independent count of tape-lengths. The temperature 
of the tape must be taken every hour in the day. Stations 
should be made at even tape-lengths whenever practicable. 

8. Observations for azimuth must be made at close of 
each day’s work when possible, and azimuth stations should 
not be more than ten miles apart, except on long tangents. 

9. An azimuth observation must consist of not less than 
three direct and three reverse measures on three parts of the 
circle between Polaris and an azimuth mark, to be made at 
any hour, but preferably near elongation, and the place, date, 
time, and watch error should be recorded. 

10. The watch should be compared with standard time 
often enough to determine its error within ten seconds. 

11. Where the line traversed is very crooked the instru¬ 
ment should be fitted for observation of solar azimuths, and 
these should be made at least twice in each day, weather per¬ 
mitting, in addition to Polaris observations. 

12. The record must contain a description of the starting- 
point of the line and the beginning and ending of each day’s 
work; also, location of each railroad station, mile-post, and 
switch passed, and wagon-road, stream, land or county line 
crossed, and connection with corners of the public-land sur¬ 
veys. 

13. Two permanent marks, either the copper bolts or the 
standard bronze tablets of the Survey, should be placed not 
less than 500 feet apart at the beginning and end of each line, 
also at prominent junction points from which other primary 
control lines may be started. A complete description and 
detailed sketch of these should be entered in the note-book. 

14. Permanent marks of some kind should be left at such 
points passed during cloudy or unfavorable weather as it may 
be necessary to return to for the observation of azimuths. 


536 FIELD-WORK OF PRIMARY TRAVERSE. 

15. Meridian marks, consisting of two of the standard bronze 
tablets let into dressed stone or masonry posts and placed 500 
feet or more apart, must be established at each county seat passed 
in the progress of the work. 

16. Observations for magnetic declination must be made 
at several points in the course of a season’s work, especially 
at county seats. 

17. A complete record must be kept by the transitman in 
book No. 9-905, and a separate record of tape-lengths by the 
front tapeman. 

18. No primary traverse' line can be accepted until checked, 
either by completing a -circuit, or by connecting with one or more 
accurately located points (astronomic, triangulation, or traverse). 

19. Permanent marks, preferably the standard bench-mark 
posts, must be set at intervals not exceeding 8 miles; especially 
should one be set at prominent junction points from which other 
primary control lines may be started. 

20. On lines for the control of sheet borders such a mark 
should be set near the corner of each quadrangle and midway 
between the corners, all such marks being turning points in the 
instrument line. 

21. If level bench-marks have already been established along 
the route of survey, they should be tied to, and may thus serve 
as permanent marks. 

22. Write in original notebook full and definite descrip¬ 
tions of permanent marks and of other points for which posi¬ 
tions are to be -computed, together with a sketch, giving dis¬ 
tance in feet to the nearest object which can be identified, such 
as mile-posts, schoolhouses, churches, ferries, bridges, farm¬ 
houses with names, fences, gates, prominent trees, etc. 

23. Trees, fence-posts, etc., along the traverse line should 
either be blazed or painted, so that the topographer will have 
no difficulty in following the route. 

24. The transit notes should be entered and worked up in 
the following manner: 


COST ; SPEED , accuracy. 


537 


Station. 

Pointings: Back and 
Transited. 

Mean 

Pointing. 

Deflection 

Angle. 

Comp. az. 

187° 56' 40 " = 

Remarks. 

= Observed Azimuth 

Ver. A. 

Ver. B. 

392 

65° 4 8' 00" 

245°48'oo ,/ 

65° 48' oo" 

o° 46' oo" 


II A.M. 72 0 . 


65 02 00 

245 02 OO 

65 02 00 




100 rails. 



65 02 00 

0 46 00 

187 10 40 

76 road crossing. 

2,975 

64 16 00 

244 t6 00 

64 16 00 








0 46 00 

— 06 

(Corr. per station) 

393 

64 16 00 

244 16 00 

64 16 00 

4 30 

187 10 34 

12 m. 73 0 . 


68 57 30 

248 57 30 

68 57 30 




39 rails. 



68 57 30 

4 4 1 3 ° 

igi 52 IO 

On road crossing. 

1,197 ft - 

73 39 00 

253 39 oo 

73 39 co 


— 12 






4 4 i 3 ° 

— 







191 51 58 



232. Cost, Speed, and Accuracy of Primary Traverse. 

—Primary traverse executed by the U. S. Geological Survey 
costs from three to five dollars per linear mile, according to 
the topography of the country, and has averaged generally 
about $3.50 per linear mile. The speed made varies between 
two and ten miles per day, also depending upon the topog¬ 
raphy. With parties of from five to seven men the daily cost 
is from Si 5 to $25. 

The primary traverse lines of the above organization are 
from 50 to 300 miles in length, averaging 150 to 200 miles. 
The closure errors of such lines vary within a wide range, and 
there seems to be no accounting for their erratic character. 
They have been found to average from 10 to 200 feet per 
100 miles of traverse, and their probable error , therefore, 
varies between 1 : 3000 and 1 : 50,000. Where the topo¬ 
graphic control is to be platted on a scale of 1 mile to 1 inch, 
it will thus be seen that the error in a primary traverse of 100 
miles length may be a perceptible quantity if the error be in 
excess of one foot in a mile. Ordinarily in a distance of 100 
miles the error is so small that it can be practically eliminated 
by the adjustment to the points which control the extremities. 
(Art. 226.) 





















CHAPTER XXIV. 


COMPUTATION OF PRIMARY TRAVERSE. 

233. Computation of Primary Traverse. —The compu¬ 
tation of the primary traverse involves: 

1. Correction of tape-lengths for temperature; 

2. Correction of tape-lengths for inclination; 

3. Reduction of measured distance to sea-level; 

4. Determination of mean angle; 

5. Computation of deflection angle; 

6. Correction to recjuce to observed astronomic azimuths; 

7. Computation of latitudes and longitudes of controlling 
points on the traverse; and 

8. Correction from adjustment to check astronomic posi¬ 
tions. 

The correction of tape-lengths for temperature and inclina¬ 
tion are explained in Articles 216 and 219, as is the mode of 
reducing the distances to sea-level in Article 220. The 
derivation of the mean deflection angle is clearly indicated in 
the example given in Article 230, as is the method of com¬ 
puting and correcting the azimuth. 

Under ordinary circumstances the temperature correction 
is a negligible quantity, as it is far less in amount than the 
other avoidable errors. So also is the correction for inclina¬ 
tion , providing the tape is held horizontal as it should be. 

The computed azimuths are corrected by the nightly 
azimuth observations, the new astronomic azimuth being 

538 


CORRECTION FOR OBSERVED CHECK AZIMUTHS , 539 


adopted in place of that brought forward. In case of a small 
discrepancy between the two such correction should be uni¬ 
formly distributed between two consecutive azimuth stations 
(Art. 309). The above corrections having been made, latitudes 
and departures may now be computed (Arts. 90 and 235) for 
each station commencing at the initial point or that for which 
geodetic coordinates have been previously obtained. Such 
latitudes and departures are computed one from the other, 
dimensions being in feet, the sum of latitudes being converted 
into seconds to give differences in latitude, and the sum of 
departures into seconds to give seconds in longitude. 

234. Correction for Observed Check Azimuths. —The 
method of correcting the computed azimuths by the observed 
check azimuths is illustrated in the example in Art. 230. 
Azimuth was observed at Traskwood between stations 133 
and 134, and was found to be 42 0 55' 15". The azimuth 
brought forward from the last azimuth station, 26 instrument 
stations distant, was 42 0 58' 07". The difference between 
the observed and the computed azimuth at Traskwood was 
2' 52". There were accordingly 172" to be distributed be¬ 
tween the 26 stations, or, giving the proper algebraic signs, 
the correction amounted to 06''.5 per station. 

The convergence of meridians subtracted from the apparent 
error in azimuth 2' 52", gives the actual error of the computed 
azimuth. Table XXX gives the convergence of meridians for 
every six miles. As the distance in longitude in the above ex¬ 
ample was three miles, the convergence for the mean latitude, 
34 0 30', amounts to C 45". This from the apparent error 
2' 52" shows the actual azimuth error to be C 07". 

Convergence of meridians, which is the amount by which 
they approach from their greatest distance apart at the equator 
until they intersect at the pole, may be determined approxb 
mately by the rule: “A change of longitude of one degree 
changes the azimuths of the straight line by as many minutes 
as there are degrees from the latitude of the place.” 


540 


COMPUTATION OF PRIMARY TRAVERSE. 


Table XXX. 

CONVERGENCE OF MERIDIANS SIX MILES LONG AND SIX 

MILES APART. 


(From U. S. Land Survey Manual.) 


Latitude. 

Convergence. 

Difference of Longitude 
per Range. 

Longitude. 

Difference 
of Latitude 
for 1 Mile 
in Arc. 

On the 
Parallel. 

Angle. 

In Arc. 

In Time. 

Arc of . 

O 

Feet. 

/ 

// 


// 

Seconds. 

Feet. 



30 

24.7 

3 

0 

6 

0.36 

24.02 

87.9 



31 

28.8 

3 

7 

6 

4.02 

24.27 

87.1 



32 

30.0 

3 

15 

6 

7.93 

24-53 

86.1 


. 

1 o'.871 

33 

31.2 

3 

23 

6 

12.00 

24.80 

85.1 



34 

32.4 

3 

30 

6 

16.31 

25.09 

84.2 

- 


35 

33-6 

3 

38 

6 

20.95 

25.40 

83.2 



36 

34-8 

3 

46 

6 

25.60 

25-71 

82.2 



37 

36-1 

3 

55 

6 

30.59 

26.04 

81.1 


!* 0/870 

38 

37-5 

4 

4 

6 

35.8i 

26.39 

80.1 



39 

38.8 

4 

13 

6 

4L34 

26.76 

78.9 

J 


40 

40.2 

4 

22 

6 

47-13 

27.14 

77-8 



4i 

41.6 

4 

3i 

6 

53-22 

27-55 

76.7 



42 

43-2 

4 

4i 

6 

59.62 

27.97 

75-5 


*0/869 

43 

44-7 

4 

5i 

7 

6.27 

28.42 

74-3 



44 

46.3 

5 

1 

7 

13 44 

28.90 

73-1 

- 


45 

47-9 

5 

12 

7 

20.93 

29-39 

71.9 

1 


46 

49.6 

5 

23 

7 

28.81 

29.92 

70.6 



47 

5L3 

5 

34 

7 

37.10 

30.47 

69-3 

}> 0/869 

48 

53-2 

5 

46 

7 

45-79 

31.05 

68.0 



49 

55-i 

5 

59 

7 

55-12 

31.67 

66.7 

J 


50 

57-i 

6 

12 

8 

4.90 

32.33 

65-3 


0/868 


235. Computation of Latitudes and Longitudes. —The 

following is an example of the form employed in the United 
States Geological Survey in computing the differential lati¬ 
tudes and longitudes of the several traverse stations. This 
is almost identical with the method already described for 
computing latitudes and departures (Art. 90). 

While the form here used is apparently complicated, it is 
in reality very simple and condensed. Instead of the loga- 


























COMPUTATION OF LATITUDES AND LONGITUDES. 54I 

rithms of the distances being arranged as separate columns 
for addition to the sines and cosines of the azimuth, to obtain 
respectively the logarithms of distances in longitude and lati¬ 
tude, they are arranged here in one column. When the loga¬ 
rithm of cosine is to be added to the logarithm of distance 
the logarithm of the sine is covered by a lead-pencil or slip of 


Station. 

Distance 
and Azimuth. 

Logarithms 


107 to 108 

1800 

3-25527 

Log of dist. 


44° 30' IO" 

9.84568 

Sine of az. 



9.85322 

Cosine of az. 


South 128 If ft. 

3 . 10 S 49 

Log of dist. + log cos az. 


West 1262 “ 

3.10095 

Log of dist. -f- log sine az. 

108 to IO9 

3000 

44 0 30' 04" 

3-47712 

Log of dist. 


9.84567 

Sine of az. 



9.85323 

Cosine of az. 


South 2140 ft. 

3.33035 

Log of dist. -f- log cos az. 

— 

West 2103 ‘ ‘ 

3.32279 

Log of dist. -f- log sine az. 


* * * * * * * * * 

* * * * * # * * # 


132 to 133 

1200 

42° 18' 45" 

3.07918 

9.82813 

9.86893 

Log of dist. 

Sine of az. 

Cosine of az. 


South 887 ft. 
West 808 “ 

2 . 9 48 11 
2.90731 

Log of dist. - 4 " log cos az. 

Log of dist. + log sine az. 

133 -f 130 ft. 

130 

42 ° 55 ' 15 " 

2.11394 
9 - 833 I 4 
9.86468 

Log of dist. 

Sine of az. 

Cosine of az. 


South 95 ft. 
West 89 “ 

1.97862 
1 . 94708 

4 

Log of dist. -J- log cos az. 

Log of dist. -f- log sine az. 

- — - ■■ 



























































542 


COMPUTATION OF PRIMARY TRAVERSE. 


paper, and the sum is placed in the fourth line, marked 
“ Logarithm Distance -f- Logarithm Cosine.” Likewise when 
the logarithms of sines and distances are added the logarithm 
of cosine is blocked out and the result put in the last line. 
The number corresponding to the logarithms in the last two 
lines, as obtained from a table of logarithms (Tables V and 
VI), is given on the left end of this line opposite the word 
“ South ” or “ West ” as the case may be, and is in feet. In 
computing primary traverse a five- or seven-place table of loga¬ 
rithms should be used as in all other primary computations. 

236. Corrected Latitudes and Longitudes. —The lati¬ 
tudes and longitudes as computed in the last article are their 
respective amounts in feet, and may be used in this form in 
platting. That these may be reduced to their geodetic 
coordinates and finally corrected by adjustment to observed 
astronomic positions (Art. 309) the computation is continued 
as follows: 

All northings and southings and all eastings and westings 
are added together, and the differences of north and south and 
of east and west are obtained by algebraic summation. In 
the following example the line was run in one direction ; and, 
accordingly, there is no sum of norths to be subtracted from a 
sum of souths, nor easts to be subtracted from wests. Thus: 


South. 

West. 

1284 

1262 

2 I4O 

2 103 

X 

X 

X 

X 

887 

808 

95 

89 


37,860 = Total South 20,644 = Total West 

The total south and west is changed to arc by adding the 
arithmetic complement of the logarithm of the value of one 




CORRECTED LATITUDES AND LONGITUDES. 543 

second of arc in meters to the constant for reducing feet to 
meters. To this sum is finally added the logarithm of the total 
distance in feet of southing to get latitude. The number corre¬ 
sponding to the sum gives the correction or change of latitude 
in seconds. The same operation is performed to obtain the 
corrections in longitudes or departures, by adding to the total 
westing in feet the logarithmic constant of feet to meters, and 
the a. c. log. value of one second in meters. Thus: 

a. c. log. value 1" in meters.. 8.51125 log. total west (Long.) 20644. 4.31479 


log. reduc. ft. to meters. .. 9.48401 log. ft. to meters. 9.48401 

log. total south (Lat.) 37860.. 4.57818 log A , Table XXXVII. 8.50926 

Lat. cor. = 374".49 (No.). 2-57344 (log-) Log- Secant L' . 0.08374 


Long, correction = 246 // .49 (No.).. 2.39180 (log) 

The logarithmic value of one secojid is obtained for the 
example by finding in Table XXIV, opposite the approximate 
latitude 34 0 30', the length in meters of one degree of arc, 
110,919. This divided by 3600" gives the value of one second, 
30.813. The logarithm of this is 1.48875, and its arithmetic 
complement is 8.5 1125. The logarithm of the constant for 
reducing feet to meters is derived from Table XLIII. 

The latitude and longitude of the last station, say Benton 
Depot, being known, those of the next station, asTraskwood 
Depot, are obtained by adding to or subtracting from the 
former according as it is north or south, i.e., plus or minus, 
the amount of change in minutes and seconds between the 
two. Thus: 


Latitude. Longitude. 

34° 33 r n ,, .92 Benton Depot (middle window). 92 0 35' 18".63 

— 6 14. 49 Correction for new position . +4 06. 49 


34° 26' 57".43 Traskwood Depot (middle window). 92 0 39' 25". 12 


If now this primary traverse has been run between an ob¬ 
served astronomic position at Little Rock and another astro¬ 
nomic position, say at Fort Smith, Arkansas, the latitudes 
and longitudes as brought through by the primary traverse 





















544 


COMPUTATION OF PRIMARY TRAVERSE. 


computation from Little Rock to Fort Smith will be in error 
by a certain number of seconds. This error may be dis¬ 
tributed by dividing its amount in seconds by the distance in 
miles, and the correction per mile applied to the various 
computed positions. Or, providing known portions of the 
traverse are for any reason more liable to be in error than 
others, arbitrary weights may be applied in distributing the 
error. A rigid adjustment by the method of least squares 
is scarcely warranted by the quality of the work and the 
number of conditions. (Art. 264.) 


CHAPTER XXV. 


C)f ; 


FIELD-WORK OF PRIMARY TRIANGULATION. 

237. Primary Triangulation.—The purpose of primary 
triangulation is the determination of the relative position upon 
the face of the earth of various commanding points. This 
operation involves a knowledge of the astronomic position of 
some initial point, as an extremity of a base line or some 
other known point; and of the distance in standard measure 
between the initial and some other intervisible point, as the 
two extremities of a base line or the imaginary line joining 
two known triangulation positions. 

The operations of triangulation involve the measurement 
of the angle at the initial point, between two intervisible 
points, the position of one of which is known ; also the angles 
at the other two points, thus giving the three angles of the 
triangle. P'iftally, with the known length of one side and the 
three measured angles of the triangle, the other sides of the 
triangle may be computed. (Art. 259.) 

Triangulation, as executed in connection with geodetic and 
topographic operations, may be divided into three kinds ac¬ 
cording to its precision : 

1. Primary triangulation, with sides varying from 15 to 
100 miles or more in length, and executed with the best in¬ 
struments in the most accurate manner; 

2. Secondary triangulation, with sides from 5 to 40 miles 
in length, executed with surveyors’ transit or with plane- 
table; and 


545 


54-6 FIELD-WORK OF PRIMARY TRIANGULATION. 


3. Tertiary triangulation, with sides less than 10 miles in 
length, executed in the progress of the second, and consist¬ 
ing of unoccupied locations or of resections from primary and 
secondary locations. 

Primary triangulation should, if possible, be expanded in 
such manner as to form quadrilateral, pentagonal, hexagonal, 
and other standard figures (Art. 238) by which combinations 
of angles may be obtained in order to strengthen the com¬ 
ponent angles in the course of the computations. These fig¬ 
ures are verified by the angles, and at intervals the precision 
of the whole scheme of triangulation is verified by the meas¬ 
urement of additional primary base lines with the accompany¬ 
ing astronomic determinations of coordinates. 

238. Reconnaissance for Primary Triangulation.—The 
term reconnaissance is generally meant to embrace all those 
investigations of a region about to be triangulated which pre¬ 
cede the actual field-work of base measurement (Art. 202) 
and the measurement of angles (Art. 252). Where the re¬ 
connaissance is preliminary to the execution of triangulation 
of the highest degree of precision, as for geodetic investiga¬ 
tions, it should be thorough and exhaustive and should de¬ 
velop every possible scheme of triangulation. 

Ordinarily a reconnaissaqce, while hastily made, should 
develop the most practical scheme of figures and afford such 
information as will add to the economy and rapidity of the 
work of angle measurement. The reconnaissance should be 
made with a view to avoiding as far as possible the necessity 
of occupying elevated structures, as observing scaffolds (Art. 
245); also the longest lines or the highest peaks, as the clouds 
which surround the latter will impede progress, while lines 
having greater length than one hundred miles invariably delay 
progress of the work. Lines of sight should be avoided 
which pass closely to the ground or to the vertical surface of 
any object, as a building, because of the liability to lateral 
refraction. In the course of the reconnaissance sites for check 


RECONNAISSANCE FOR PRIMARY TRIANGULATION. 547 

base lines should be sought every 150 to 200 miles, or, say, 
every eight or ten figures, depending on the length of the 
bases. 

In the condjict of a reconnaissance little difficulty will be 
encountered where the elevations are great and the summits 
comparatively clear of timber. If, on the other hand, the 
summits are heavily wooded and comparatively uniform in 
height, the greatest skill will be required and the slowest prog¬ 
ress made in selecting intervisible points which are most favor¬ 
ably situated for the extension of the triangulation and the 
formation of the most satisfactory figures. In the flat, com¬ 
paratively level country of the plains of Kansas, Nebraska, 
and thereabouts, triangulation may be practically laid off as on 
paper, the points selected being made intervisible by means 
of high signals (Art. 244), and the length of the sides being 
limited only by the curvature of the earth (Art. 239) and 
the height of the signals. If the same class of country is 
heavily covered with forests, the lengths of the lines will be 
governed by the labor and expense of clearing them. 

The outfit required in making a reconnaissance includes 
(1) a small theodolite, with circle reading by vernier to min¬ 
utes; (2) a compass-needle attached to the theodolite for 
determining the magnetic direction; (3) an aneroid barometer; 
(4) a ioo-foot steel tape; (5) a prismatic compass; and (6) a 
protractor, scale, and paper for platting the reconnaissance 
triangulation as it progresses, in order that the position of 
the points sighted may be approximately ascertained. Also 
the best available existing maps of the country, and only such 
camp outfit (Chap. XXXVIII) and assistants as will permit of 
the execution of the work and least impede the rate of progress. 

In making the reconnaissance it is necessary to keep stead¬ 
ily in view not only the limitations imposed by the necessity 
of selecting well-conditioned figures, but also that the most 
rigid requirements may be modified in accordance with the 
special features of the country as they are developed by the 


54 ^ FIELD-WORK OF PRIMARY TRIANGULATION . 

reconnaissance. Hence the best plan of triangulation will be 
that which not only satisfies the conditions prescribed, but 
will be most effective in its results and economic in its exe¬ 
cution. 

Among the more important of these requirements are: 

1. Assured intervisibility of stations; 

2. Selection of the higher summits; 

3. Maximum length of line within a limit of 100 miles; 

4. Angles not in excess of 120 or less than 30 degrees; 

5. The formation of the simplest and strongest figures 
practicable; 

6. The greatest area in view in order that the largest 
number of intermediate stations may be sighted ; and 

7. A consideration of the altitude to which the instrument 
must be raised in order that the visual ray may pass above 
intermediate obstacles. 

The first thing to be borne in mind in planning a triangula¬ 
tion is that the stations chosen shall form well-conditioned or 
standard figures { Art. 273), and that each triangle in the figures 
shall be as nearly equilateral as possible. To this end no 
angle should be smaller than 30 or greater than 120 degrees, 
except in quadrilaterals, where a few degrees more latitude 
may be permitted in the size of the angles. Hexagonal 
figures cover the largest areas, while quadrilaterals secure the 
greatest degree of accuracy. On the other hand hexagonal 
figures, because of the disposition of the stations, tend to 
retard linear progress and should be avoided, as should 
quadrilaterals with open diagonals when direct and not areal 
progress is sought. Quadrilaterals with observable diagonals, 
while giving the strongest figures, adapt themselves least to 
the topography and are found to be relatively difficult figures. 
Pentagons and quadrilaterals with central stations readily con¬ 
form to the configuration of the country however complex 
or difficult, and give figures of considerable strength. 

In the course of the reconnaissance extensive notes should 


INTERVISIBILITY OF TRIANGULATION STATIONS. S 49 


be made of the character of the country, the difficulties of 
travel, facilities for transportation, methods of climbing the 
various summits, the routes to them, stopping-places, etc. 
Horizontal and vertical angles should be taken on all promi¬ 
nent peaks and objects, even though it is not expected to in¬ 
clude them in the triangulation scheme. The reconnaissance 
scheme should be kept platted up each day to scale in order 
to facilitate the finding of such stations as have already been 
selected, by determining the angles to them from known points 
and to thus aid in their recognition from new stations. 

239. Intervisibility of Triangulation Stations. —The fol¬ 
lowing table, prepared by Mr. R. D. Cutts of the U. S. Coast 
Survey, is of use in reconnaissance in deciding upon the 
height of signals and observing scaffolds to be erected. The 
line of sight from the telescope to the signal should never 

Table XXXI. 

DIFFERENCE IN HEIGHT BETWEEN THE APPARENT AND 

TRUE LEVEL. 


Distance, miles. 

Difference in Feet for— 

Distance, miles. 

Difference in Feet for— 

Distance, miles. 

Difference in Feet for— 

Curva¬ 

ture. 

Refrac¬ 

tion. 

Curva¬ 
ture 
and Re¬ 
fraction 

Curva¬ 

ture. 

Refrac¬ 

tion. 

Curva¬ 
ture 
and Re¬ 
fraction 

Curva¬ 

ture. 

Refrac¬ 

tion. 

Curva¬ 
ture 
and Re¬ 
fraction 

I 

0.7 

O. I 

0.6 

23 

353 -o 

49 4 

303-6 

45 

1351-2 

189.2 

1162.0 

2 

2.7 

0.4 

2.3 

24 

3 8 4-3 

53-8 

33 ° -5 

46 

14H.9 

197.7 

I2I4.2 

3 

6.0 

0.8 

5-2 

25 

417.0 

58-4 

358.6 

47 J 

1474.0 

207.3 

1267.7 

4 

10.7 

i -5 

9.2 

26 

45 1 • 1 

63.1 

388.0 

48 

1537-3 

215.2 

1322.1 

5 

16.7 

2-3 

14.4 

27 

486.4 

68.1 

418.3 

49 

1602.0 

224.3 

1377.7 

6 

24.0 

3-4 

20.6 

28 

523 -1 

73-2 

449.9 

50 

1668.1 

233-5 

1434-6 

7 

32-7 

4.6 

28.1 

29 

561.2 

78.6 

482.6 

51 

1735 5 

243.0 

13925 

8 

42.7 

6.0 

36.7 

30 

600.5 

84.1 

516.4 

52 

1804.2 

252.6 

1551-6 

9 

54 -o 

7.6 

46.4 

31 

641.2 

89 8 

55 i -4 

53 

1874-3 

262.4 

1611.9 

10 

66.7 

9-3 

57-4 

32 

683.3 

95-7 

587.6 

54 

1945-7 

272.4 

i 673-3 

I I 

80.7 

H -3 

69.4 

33 

726.6 

101.7 

624.9 

55 

2018.4 

282.6 

1735-8 

12 

96.1 

! 3-4 

82.7 

34 

771-3 

108.0 

663.3 

56 

2092.5 

292.9 

1799.6 

>3 

112.8 

15-8 

97.0 

35 

817.4 

114.4 

703-0 

57 

2167.9 

303-5 

1864.4 

M 

130.8 

18.3 

112.5 

3 6 

864.8 

121.1 

743-7 

58 

2244.6 

314.2 

1930.4 

1 5 

150.1 

21 .O 

129. I 

37 

9 ' 3-5 

127.9 

785.6 

59 

2322.7 

325-2 

1997-5 

16 

170.8 

23 9 

146.0 

38 

9 ^ 3-5 

134.9 

828.6 

60 

2402.1 

336.3 

2065•8 

17 

192.8 

27.O 

165.8 

39 

1014.9 

142.1 

872.8 

61 

2482.8 

347 6 

2135.2 

t8 

216.2 

30.3 

185.9 

40 

1467.6 

149-5 

918.1 

62 

2564.9 

359 -i 

2205.8 

*9 

240.9 

33-7 

207 2 

41 

1121.7 

157-0 

964.7 

63 

2648.3 

370.8 

2277.5 

20 

266.9 

37-4 

229. s 

42 

1177.0 

164.8 

1012.2 

64 

2733 0 

382.6 

2350.4 

21 

294-3 

41.2 

253.1 

43 

1233.7 

172.7 

1061.0 

65 

2819.1 

394-7 

2424.4 

22 

322.9 

45 • 1 

277.7 

44 

1291.8 

180.8 

11 I I .O 

66 

2906.5 

406.9 

2499.6 

































550 FIELD-WORK OF PRIMARY TRIANGULATION. 


pass less than 6 feet above the earth’s surface at the tangent 
point, and should be higher, if possible, to reduce errors from 
unequal refraction. 


Curvature = 


square of distance K 2 

mean diameter of earth — 2R’ 



Log curvature = log square of distance in feet — 7.6209807. 

K 3 , \ 

Refraction = —m, .(48) 

where K = the distance in feet; 

R = mean radius of the earth (log R — 7.3 199507); 
and 

m = the coefficient of refraction, assumed at .070, it& 
mean value, seacoast and interior. 


R 3 

Curvature and refraction = (1 — 2 tn)—=. 

v 2R 



Or, calling h the height in feet, and K the distance in statute 
miles, at which a line from the height h touches the horizon, 
taking into account refraction, assumed to be of the same 
value as in Table XXXI (0.70 for one mile), we have 



and h = - K ' . 

1.7426 


An approximate , yet comparatively accurate, empirical 
formula for determining the combined effect of curvature and 
refraction is 


Curvature and refraction, in feet = 0.574 (distance in miles) 3 . 

The following examples will serve to illustrate the use of 
the preceding table: 

I. Elevation of Instrument required to Overcome Curvature 
and Refraction. —Let us suppose that a line, A to B , was 
18 miles in length over a plain, and that the instrument 
could be elevated at either station, by means of a tripod. 






INTERVISIBILITY OF TRIANGULATION STATIONS. 551 

« 

to a height of 20 or 30 or 50 feet. If we determine upon 
36.7 feet at A, the tangent would strike the curve at the dis¬ 
tance represented by that height in the table, viz., 8 miles, 
leaving the curvature (decreased by the ordinary refraction) 
of 10 miles to be overcome. Opposite to 10 miles we find 
57.4 feet, and a signal at that height erected at B would, 
under favorable refraction, be just visible from the top of the 
tripod at A, or be on the same apparent level. If we now 
add 8 feet to tripod and 8 feet to signal-pole, the visual ray 
would certainly pass 6 feet above the tangent point, and 20 
feet of the pole would be visible from A. 

II. Elevations required at given distances . — If it is desired 
to ascertain whether two points in the reconnaissance, esti¬ 
mated to be 44 miles apart, would be visible one from the 
other, the natural elevations must be at least 278 feet above 
mean tide, or one 230 feet, and the other 331 feet, etc. This 
supposes that the intervening country is low, and that the 
ground at the tangent point is not above the mean surface of 
the sphere. If the height of the ground at this point should 
be 200 feet above mean tide, then the natural elevations 
should be 478, or 430, and 531 feet, etc., in height, and the 
line barely possible. To insure success, the theodolite must 
be elevated, and at both stations, to avoid high signals. 

III. To determine ■whether the line of sight between tzvo 
stations would pass above or belozv the summit of an intervening 
hill , and how much in either case . (Fig. 160.) 



Fig. 160.—Intervisibility of Objects. 


hi = height of lower station. d\ = distance h x to hi. 

hi = height of higher station. d% — distance hi to k%. 

hi = height of intervening hill. 






55 2 FIELD-WORK OF PRIMARY TRIANGULATION. 


Example I. 


h, 

h, 

hi 

d l 


600 feet. 
2000 feet. 
1340 feet. 
54 miles. 


di — 10 miles. 


600 feet strikes horizon at 32.3 miles, 
64 — 32.3 = 31.7 miles = 577 feet, 

31.7 — 10 = 21.7 miles = 270 feet, 

2000 = 577 feet = 1423 feet, 


64 1423 

— = 0.4 and —— 
10 6.4 


222.3 feet, 


and hi, or height of line at 7 / 3 = 1423 -f- 270 — 222.3 = 1470 feet. 


Hence the line passes 130.7 feet above the intervening 
hill and the stations are intervisible. 

Example IE 


hi — 900 feet. 
hi = 3600 feet. 
hi — 1980 feet. 
di = 55 miles. 

di — 25 miles. 

and h = 2654 + 139*8 


900 feet strikes horizon at 
80 miles — 39.4 miles = 40.6 miles 


40.6 — 25.0 

3600 feet — 946 feet 

80 , 2654 

— =3.2 and -- 

25 3-2 

829.4 = I964.4.feet. 


= 15.6 miles 


39.4 miles, 
= 946.0 feet, 
= 139.8 feet, 
= 2654.0 feet, 

= 829.4 feet, 


Hence the summit at h 9 is 15.6 feet higher than the line 
of sight, and the two stations are not intervisible. 

If we elevate the instrument 60 feet at h % , the line would 
pass clear of /z 2 , or its height at that point would be 2006 
feet. 

The question of intervisibility may be also determined by 
the following formula, in which the coefficient of refraction 
is reduced to .065 : 


h — h x -f- — //,) 


d , 


0.5803 d x d % . 



Example III. Same data employed as in Example I. 


[hi — hi) = 1400 feet. 

<L — 54 miles. 

d y di = 64 miles. 


logs. 

log (7/3 — hi) = 3.14613 di = 54 miles 1.73239 
log di = 1.73239 di = 10 miles 1.00000 

Co. log (di di ) = 8.19382 Constant 9.76365 


1181.2 feet 


3.07234 313.4 feet 2.49604 


and hence h =600 feet + 1181.2 — 313.4 = 1467.8 feet- 









ACCURACY OF TRIANGULA TION. 


553 


240. Accuracy of Triangulation.—The precision of a 

scheme of primary triangulation is dependent upon several 
related quantities; notably — 

1. The precision of the astronomic determination of the 
geodetic coordinates of the initial point. 

2. The precision with which the base line on which the 
triangulation is dependent has been measured. 

3. The care taken in gradually transferring the short length 
of a base through expansion to the longer sight lines of the 
triangles. 

4. The errors inherent in the measurement of the ang-les 

o 

of the triangulation, including those due to instrument is well 
as signal. 

The probable error of an astronomic determination (Art. 
327) is considerably greater than that of the measure of a base 
line and nearly as great as that introduced within the expan¬ 
sion of the triangulation. A base line can readily be meas¬ 
ured with a probable error far less than that which can be 
maintained in the execution of the triangulation (Art. 213). 
It is impossible in triangulation to maintain an accuracy at 
all approaching that of the base measured with a probable 
error of ¥Tnr ^ Tnnr , while an accuracy of even ^Trow is difficult 
to maintain in an extended triangulation. The accuracy of 
the base is lost partly in the base figure and is rapidly dissi¬ 
pated in the adjacent expansion. The figures used in ex¬ 
panding a base line through the net of triangles to the outer 
triangles are, therefore, of great importance. Ideally they 
should be a series of quadrilaterals with diagonals intersect¬ 
ing at right angles. The lengths of their sides should be 
increased in a ratio of 1 to 2 or 3, thus requiring two or 
three steps or series of figures in the expansion to reach the 
outer triangulation scheme. (Fig* 161). 

241. Instruments.—The various tools employed in the 
measurement of angles of a scheme of primary triangulation 
may be classed under two general heads: 



554 FIELD-WORK OF PRIMARY TRIANG U LA PI OK. 


1. Instruments for measuring horizontal angles; and 

2. Signals or objects upon which to observe. 

There is practically only one form of instrument em¬ 
ployed in the measurement of horizontal angles, and this is 
known as a theodolite. In its general characteristics it re¬ 
sembles an engineer’s transit (Art. 85), from which it differs 
chiefly— 

1. In the fact that the telescope does not transit or re¬ 
volve vertically through 180 degrees in the wyes; 

2. In the special care exercised in making the instrument, 
particularly in the accuracy of centering and fitting the ver¬ 
tical axis and strict uniformity of graduation ; and 

3. In the size of the telescope and horizontal circle, and 
the mode of reading the graduations upon the latter. 

The earlier theodolites were so constructed that the circle 
was read with verniers , and in order that a proper degree of 
precision might be attained they were made excessively large, 
having circles from 16 to 30 inches in diameter. Experience 
has proven, however, that with the aid of micrometer micro¬ 
scopes (Art. 242) greater accuracy of measurement can be had 
by use of an instrument having a horizontal limb not exceed¬ 
ing 12 inches in diameter; while work which is sufficiently 
accurate for all the purposes of an ordinary triangulation 
can be executed with theodolites having circles of 8 inches 
diameter. 

In Fig. 161 are indicated two typical expansions of the 
base , AB, by means of nearly ideal figures. The first is by 
enlargement to the quadrilateral ACBD; the second would 
be employed when signals could not be observed in the 
direction C, and would be by expansion to the quadrilateral 
ABDF. An especially strong pentagonal figure is formed 
when all these directions can be sighted. From these expan¬ 
sion may be made to the still longer sides of the quadrilateral 
CHIF , or of the pentagonal figure GHIFE. 


INS TR UMEN TS. 


555 


Where superior theodolites are employed it has been 
found that the probable error of the measure of a direction 
may vary between i and 5 seconds of arc. Some of the 
best work of the U. S. Coast Survey has given results vary¬ 
ing between .6" and .75". Any triangulation in which this 
does not exceed 1" may be classed as of the highest order. 



Measures attaining accuracy of from 2 to 5 seconds are suf¬ 
ficiently refined for all the ordinary purposes of a primary tri¬ 
angulation outside of those required in geodetic investiga¬ 
tions. 

The latest direction theodolites used by the U. S. Coast 
Survey have circles of 12 inches diameter with double centers, 
the outer center of cast iron, and the inner of hardened steel. 
The inner center and socket are made with great precision ; 
the outer center and socket are well made, though constructed 
with less precision, as this center serves only for shifting the 
position of the circle and not for the reading of angles. The 
alidade, by which is meant everything above and including 
the graduated circle, the wyes, and the telescope, is supported 




55^ FjFLD-WORK OF PRIMARY TRIANG U RATION. 

on the inner center and is made of aluminum as far as prac¬ 
ticable, and so constructed that the friction upon the center 
is exceedingly small. The center is 8 inches long, its bear¬ 
ing surfaces being cones. The circle is covered to protect 
it from dust, and is made especially heavy and the centers 
long, to give stability. In spite of this the weight is but 
41 pounds, due to the extensive use of aluminum. The 
circle is graduated on coin silver and is divided to five min¬ 
utes, and is read to two seconds of arc by means of three 
equidistant micrometer microscopes. Each degree of the 
graduation is numbered. The telescope objective is 2.4 
inches aperture and 29 inches focal length. A striding-level 
which rests on the axis supporting the telescope is graduated 
to 4 seconds of arc. 

The direction theodolites used in the U. S . Geological Sur¬ 
vey are supported on heavy split-leg wooden tripods and rest 
upon aluminum tripod heads. (Fig. 162.) The circles of 
these instruments have a diameter of 8 inches and are divided 
to 10 minutes, though they can be read to 2 seconds of arc 
by means of two micrometer microscopes placed on opposite 
sides of the alidade. The object-glass is 2 inches in diameter 
and has a focal length of i6f inches, with an eyepiece having 
a magnifying power of about 30 diameters. 

242. Micrometer Microscope. —This is a device for the 
measurement of smaller parts of an arc than are indicated by 
the graduations upon it. Micrometer microscopes are used 
in high-power angle-reading instruments in place of the ver¬ 
niers used on engineering instruments, since they give more 
accurate results and finer subdivisions of the arc. They con¬ 
sist of a microscope generally supported upon the standards 
of a theodolite with the objective end in close contact with 
the horizontal circle, and lighted by a cylindrical glass ex¬ 
tension. It is sometimes called the filar micrometer , because 
the small measurements are made by means of fine threads 



THEODOLITE, 


557 





































































































































































55$ FIELD-WORK OF PRIMARY T RIANG U LA TION. 


or films. These are similar to the cross-hairs of the eye¬ 
piece of a telescope. The instrument consists of three sepa¬ 
rate parts: 

1. The microscope tube, carrying the lenses for magnify¬ 
ing the divisions on the circle and the hairs; 

2. A large-headed screw the outer circumference of which 
is divided, and is read by means of a fixed pointer; and 

3. A comb-scale and cross-hairs by which the divisions 
of the circle are read and subdivided. 

The micrometer cross-hairs and comb-scale are fixed in the 
plane of the image produced by the objective of the micro- 



'Fig. 163. —Section of Micrometer through Screw showing Comb 
and Cross-hairs in Central Plan. 


scope. This image is larger than the object seen in the 
microscope, therefore a given amount of the micrometer 
cross-hairs corresponds to a much less distance on the ob¬ 
ject sighted. The cross-hairs are held in a frame which is 
moved by a screw having a very fine thread, called the mi¬ 
crometer screw . (Fig. 163.) This is caused to revolve by 
a large head, called the micrometer head , which is cylin¬ 
drical or hollow, its outer circumference being divided into 
-jsixty. 

The relation between the comb-scale of the microscope, 
and the graduations on the micrometer head which denote 
the fractions of a revolution of the screw, is such that one 
full revolution of the screw corresponds to one tooth of the 





































MICRO ME TER MICROSCOPE. 


559 


comb-scale. The number of whole revolutions of the screw 
are recorded by noting how many teeth of the comb-scale 
are passed over; the fractional parts of a revolution being 
read on the graduated micrometer head. 

If the circle of the instrument is divided to 10 minutes 
and the micrometer read to 2 seconds, as in the case of the 
8 -inch theodolite of the U. S. Geological Survey, the heads of 
the micrometers are divided into sixty parts numbered o, io, 
20, o, io, 20. One revolution of the micrometer screw is 
equivalent to 2 minutes, and one division of the head to 
2 seconds. The comb-scale of the micrometer consists of ten 
parts, each of which corresponds to the space of io minutes 
on the circle and to five revolutions of the micrometer screw. 
The slide by which it is read carries two cross-hairs close 
together. The micrometer records zero near the middle 
when one of the two cross-hairs is on the middle line of the 
comb-scale and the head of the screw is at zero. Degrees 
and minutes are read directly in the microscope. The read¬ 
ings of the micrometer head are recorded in the notes as 
divisions. The sum of the readings of the heads of the two 
separate micrometer microscopes gives the mean reading of 
the two in seconds. 

Five revolutions of the screw should move the cross-hairs 
from one graduation to the next. If this is not exactly true, 
then the value of the ten-minute space should be measured a 
number of times by running the cross-hairs backward and for¬ 
ward. The mean of these five revolutions should give the 
mean value of one revolution of the micrometer screw, and 
this is called the run of the screw . When reading the instru¬ 
ment a correction is to be applied called the correction for 
run, and this is determined as described above for various 
parts of the micrometer screw. (Art. 251.) 

243. Triangulation Signals. —There are three general 
forms of signals upon which to observe or point the cross- 


5^0 FIELD-WORK OF PRIMARY TRIANG ULATION. 


hairs of the telescope in the measurement of angles of a pri¬ 
mary triangulation. These are: 

1. Opaque signals, usually tripods or poles of wood with 
flag or other opaque device attached thereto; 

2. Reflecting signals ; and 

3. Lights or night-signals. 

Opaque signals should generally be employed where the 
conditions of the atmosphere and the lengths of the sights 
will permit. A smaller probable error results from observing 
upon them than upon any other form of signal. 

Reflecting signals are of two general types: 

1. Tin reflecting cones or other stationary objects of 
conical or cylindrical shape ; and 

2. Heliotropes, or instruments by which sunlight is re¬ 
flected by a mirror towards the observer. 

Neither heliotropes nor tin reflecting cones permit of as 
accurate results in observing as do opaque signals, because of 
the phase or displacement of the reflected beam of light, 
which is often considerable. The most satisfactory reflecting 
signal upon which to observe, because of the certainty of its 
being seen in hazy and foggy weather or on timber-covered 
summits, is the heliotrope. The flash of the reflected sunlight 
from this instrument can be seen from the most distant points 
which can be observed, as well as on those partially obscured 
by atmospheric conditions. 

In smoky and hazy weather the atmosphere is clearest at 
night , and it may be necessary to use reflectors illuminated by 
ordinary kerosene lamps or, on very long lines, by magnesium 
tape burned in and reflected by a special apparatus. 

The correction for phase in tin cones , or reduction to the 
center of the signal, is 


Cor. = 


r cos 2 1 Z 
± D sin i 7 "" 



TRIPOD AND QUADRIPOD SIGNALS. 561 

in which r = radius of signal,- 

Z = angle at point of observation between the sun 
and the signal, and 

D — distance from observer to signal. 

244. Tripod and Quadripod Signals.—Various forms of 
opaque signals are employed, according to the length of sight 
and the availability of materials for construction. Where 
the circumstances will permit, the simplest form is a tripod 
from the center of and above which projects a pole carrying 
cross-pieces to which are fastened strips of cloth at right 
angles in target form, while the whole may be surmounted 
by a flag. (Fig. 164.) The object of the flag is that, in 
waving in the sunlight, its white flash is more readily dis¬ 
tinguished and more quickly attracts the observer than do 
the stationary tripod and targets. This form of signal must 
be accurately centered over the station mark, the bottom of 
the pole being cut off so that the theodolite can be set 
beneath the tripod and over the station mark. 

The various details of the tripod signal, such as— 

1. Height of pole ; 

2. Dimensions of the tripod; 

3. Boarding-in of the lower part or covering it with can¬ 
vas to shade the instrument and protect it from high winds; 

4. Whether the pole or the cloth upon it shall be white 
or black; 

5. Character of the signal to be employed; 
and other matters of detail vary with— 

a. Length of the line observed; 

b. Altitude of the station; 

c. Background of the station ; and 

d. Atmospheric conditions encountered. 

The diameter of the signal-pole must not be greater than 
will just permit of its being distinctly seen, so that it may be 
accurately bisected by the vertical cross-hair. The latter 



562 flELD-WORK OF PRIMARY TRIANG U LAP ION. 


must not cover the pole, but must permit a portion of it to 
show on either side of the hair. The diameter which, with 
averaged-sized cross-hair and magnifying power, subtends an 
angle of 1 second at 1 mile is .307 of an inch; hence at 20 
miles it is 6.1 inches, at 40 miles it is 12.3 inches, at 60 miles 
it is 18.4 inches, and at 80 miles it is 24.6 inches. The 
above proportions show that for lines exceeding 16 miles the 
diameter of the signal should not exceed one second in value. 
The solid part of the pole should never be greater than about 
4 to 6 inches in diameter, in order that it may not be too 
heavy to raise. Its visible dimensions may be increased by 
nailing slats of wood upon it and covering these with cloth or 
with a reflecting cone of metal. 

Such signals may be constructed of any material which is 
convenient to hand, as poles cut in the woods, old rails, etc. 
It is preferable, however, to build them of 2 X 4 or 3 x 4 
sawed scantling, and this should be procured in 12- to 16-foot 
lengths, according to the height to which it is necessary to 
raise the flag above the ground surface in order that it may 
be seen over intervening obstructions. Moreover, the upper 
portions of such a signal should be painted white. The addi¬ 
tional expense incurred in using such materials and in paint¬ 
ing will be more than counterbalanced by the added immunity 
from destruction by vandalism, a well-built and attractive- 
looking signal being far less liable to such injury than one 
crudely put together. Moreover, squared scantling can be 
more easily cut to abut against the central pole, or to be 
pieced together where it is necessary to have a signal of 
greater height than the average length of the scantling. The 
spread of the legs should be about two-thirds of the height 
of the pyramid to give stability. 

To anchor the signal , holes should be dug about 2 feet in 
depth into which the legs of the scantling should be sunk. 
Stakes 4 feet in length at least should then be driven into the 


TRIPOD SIGNALS. 


563 


I 


I 



Fig 164.— Quadrii’OD Signal 




















OB SEA VINO SCA FFOL D S. 


565 


ground in contact with the foot of each pole and approxi¬ 
mately at right angles to them, and these should be nailed to 
the scantling. This will practically insure the signal against 
being blown over. 

245. Observing Scaffolds. —Where it becomes necessary 
to elevate the instrument, a wooden scaffold must be erected, 
and this must be so constructed that the instrument can rest 
on a central scaffold entirely independent of the platform 
upon which the observer stands, in order to avoid jarring the 
instrument. The inner scaffold for the support of the instru¬ 
ment should be a tripod triangular in cross-section, and the 
inclination of its sides should be such as to give it rigidity and 
to bring the main frames together in one cap-piece at the 
summit on which the instrument will rest. The outer scaffold 
for the support of the platform on which the observer will 
stand should be square in plan, and the sides should be in¬ 
clined so as to give stability. The width of the platform at 
the top should be sufficiently great to permit of the free 
movement of the observer about the instrument, and he 
should be protected by a railing around the outer edge of the 
platform. (Fig. 165.) 

Such scaffolds may be constructed of the crudest material 
at hand, and it may be necessary often to use such only. On 
inaccessible mountains the writer has erected scaffolds in the 
tops of trees, and a central tree has been used as an observing 
stand for the instrument. In such cases all limbs should be 
cut off the tree as well as the top, in order to offer the least 
obstruction to the wind, which will otherwise jar the instru¬ 
ment. In other instances satisfactory observing scaffolds have 
been built in the limbs of a single large tree, and another 
growing close to it has been used as the observing stand. 
Wherever sawed scantling is available, however, it should be 
employed, for the reasons given in the last article. The 
scaffold should be erected much as is the framework of a 


566 FIELD-WORK OF PRIMARY TRIANGULATION. 

building. Each length should be framed on the ground until 
as many bents have been fastened together as can be readily 
raised with the force available, perhaps two bents of 8 or 12 
feet each. Those two which are opposite should be erected 
at the same time, and then the cross-bracing be nailed to place 
them at the proper distance. Thereafter boards are laid 
across the tops of each set of bents, and the workmen, stand¬ 
ing on these, frame the next higher bent. 

The whole should be strengthened by diagonal bracing 
with one-inch planks. It should be anchored as described 
in the last article, and braced, moreover, by long planks, 
leaning against it as struts and suitably grounded, or by 
guying it with long wires to neighboring stakes or trees. 

246. Heliotrope. —This is an instrument designed to 
reflect sunlight by a mirror from the station sighted upon 
to that occupied by the observer. The beam of reflected 
light is pointed upon as on a signal. There are three specific 
objects to be aimed at in the design and use of the heliotrope: 

1. The reflecting surface should be as near the center of 
the station as possible; 

2. The method of aligning or directing the reflected beam 
toward the observer’s station should be the most precise and 
simple attainable; 

3. The method of maintaining the direction of the re¬ 
flected beam, while following the apparent movement of the 
sun, should be the simplest possible. 

There are three general types of heliotropes for the 
accomplishment of the above objects. These are : 

1. Simple hand-mirrors provided with screw for attach¬ 
ment to a wooden support; 

2. Telescopes carrying revolving mirror and aligning 
sights; 

3. Steinheil heliotrope having mirror aligned by the re¬ 
flected image of the sun. 

Heliotropes should rarely be used as signals for distances 


V 


OBSERVING SCAFFOLDS. 


567 




Fig. 165. —Observing Scaffold and Signal 

































. 

' 
















■ 









































DIMENSIONS OF HELIOTROPES 


5^9 


less than twenty miles, excepting for very smoky or hazy 
weather, or because of the difficulty of making visible an 
opaque signal in dense wood ; or at any greater distance pro¬ 
viding an opaque signal can be seen. This chiefly because (1) 
opaque signal gives better definition ; (2) the beam reflected 
from the heliotrope is too large when observed at short range; 
(3) it is difficult to arrange a satisfactory understanding 
between the observer and the distant heliotroper. 

The dimensions of a heliotrope should be the smallest 
which will produce a clearly defined and visible star of light 
at the distance observed. In order, therefore, to secure 
images of uniform size at all distances, the size of the mirror 
must be varied according to the distance. For ordinary 
atmospheric conditions and distances of ten miles and over, 
the following formula may be used to determine the size of 
the mirror: 

;r = .046^/, 

in which x = the length of the sides of the mirror in inches; 
d — the distance observed in miles. 

In accordance with this formula the following table gives 
the length of the side of the mirror for various distances. 


Table XXXII. 

SIZES OF HELIOTROPE MIRRORS. 


Distance, 

Miles. 

Side, 

Inches. 

Distance, 

Miles. 

Side, 

Inches. 

Distance, 

Miles. 

Side, 

Inches. 

IO 

0.46 

60 

2.8 

120 

5-5 

20 

O.92 

70 

3-2 

140 

6.4 

30 

1-37 

80 

3-7 

160 

7-3 

40 

1.83 

90 

4 -i 

180 

8-3 

50 

2-3 

IOO 

4.6 

200 

9.2 


While the alignment of the mirror must be relatively 
precise, such accuracy is only required as may be obtained 



















570 FIELD-WORK OF PRIMARY TRIANGULATION. 


by relatively crude methods. The cone of incident and 
reflected rays subtends equal angles the amount of which 
is about 32 minutes. The base of this cone is about 50 
feet in diameter per mile of distance. Thus for a distance 
of 20 miles the reflected ray is visible over a vertical area 
of about 1000 feet diameter. It is thus evident that the 
alignment may vary as much as 15 minutes of arc on either 
side of the true direction, or nearly .01 of a foot in a distance 
of 2 feet. 

The simplest, most useful, and most practical heliotrope 
for all ordinary usage is a small hand-mirror (Fig. 166, b) sim¬ 
ilar to the reflecting mirror used with the telescopic helio- 



Fig. 166.—Telescopic Heliotrope. 


trope. The hand-mirror or the telescopic heliotrope (Fig. 
166, a) is used by inserting into the side of a tree or post 
or other wooden support the screw to which it is attached by 
means of a hinged joint working with friction. Thus, by 
screwing or unscrewing it into the wood and moving the joint, 
it can be made to follow the path of the sun. A similar 
mirror at a distance of 10 or 15 feet is used to reflect the 
sunlight into the heliotrope mirror when the position of the 
sun is such that a direct reflection cannot be cast towards the 
observing station. The reflecting or second mirror (Fig. 166, b) 














HELIO TROPES. 5 71 

is of similar construction to the hand mirror and likewise may 
be screwed into a stake, board, or tree. 

The alignment of the reflected beam from a hand mirror 
is procured by ranging the pointed tops of two small stakes 
or sticks by eye from it to the observer’s station. In order 
that this alignment may be sufficiently accurate these stakes 
must be set up at some considerable distance apart, the first 
about 10 feet and the second 20 to 50 feet distant from the 
heliotrope. Such alignment having been once made by the 
triangulator or an assistant, a heliotroper or man who shall 
move the mirror so as to keep the sunbeam on the tops of 
the two stakes may be employed, and any near-by resident 
of ordinary intelligence will be capable of performing such 
simple labor. To make sure that the reflected image is cast 
upon the tops of the stakes some dark object, as the trousers 
or hat of the heliotroper, should be placed behind them at 
intervals in order that the shadow cast by the reflected beam 
over the top of the stake may be clearly noted. Or a square 
of tin with a hole cut in it of size proportioned to the dis¬ 
tance may be held in front of the mirror as a stop. 

The telescopic or Coast Survey heliotrope (Fig. 166) con¬ 
sists of a telescope of moderate magnifying power attached to 
a screw moving with friction, by which it is fastened into a 
wooden support. Near the eye end is a mirror supported by 
a horizontal axis, and the latter may be rotated vertically so 
as to give two motions. A few inches in front of this and at 
the objective end of the telescope are two rings, so placed 
that the axes of the center of the mirror and of the rings are 
parallel to the line of sight of the telescope. The telescope 
being directed upon the observing station, the mirror is so 
turned as to reflect the sunlight through the rings and thus to 
the observing station. This instrument is less simple than 
the hand mirror because of its liability to get out of adjust - 
merit. One of the rings may be adjusted by raising or lower¬ 
ing it, and the adjustment should be frequently tested to 


5 7 2 FIELD-WORK OF PRIMARY TRIANGULATION. 


assure that the alignment of the mirror and both rings is per¬ 
fect. This operation consists in pointing the telescope at 
some object ioo or 200 feet away and noting if the reflected 
light is cast exactly upon it. If not, the adjustable ring 
must be moved to correct the error. 

The Steinheil heliotrope (Fig. 167) is a far more compact 
and serviceable instrument than the telescopic heliotrope. 
It requires, however, for its manipulation an assistant of 

some intelligence. On the other hand tele¬ 
scopic and hand heliotropes can scarcely be 
safely used by an ordinary laborer; there¬ 
fore where hand-mirrors aligned by stakes 
are not used the Steinheil will give the 
greatest satisfaction. It is but three or four 
inches in length and can be carried easily in 
the pocket or in a light leather case. This 
instrument, in addition to its portability, 
has the advantage that there are no mov¬ 
able parts to get out of adjustment by 
jarring in carrying. 

The Steinheil heliotrope consists of a 

Fig. 167.— Steinheil small sextant mirror, the two surfaces of 
Heliotrope. which are as nearly absolutely parallel as 
possible. This mirror has a small hole in the center of the 
reflecting surface, below which is a small lens in the shaft 
carrying the mirror, and below the lens is some white reflect¬ 
ing material, as plaster of Paris. The mirror is so mounted 
that it has four different motions, two about its horizontal 
axis and two about its vertical axis, each of which can be 
separately controlled by clamps or friction joints. To use the 
Steinheil , it is screwed into some wooden support, as the side 
of a tree or post, in such a position that the main axis carry¬ 
ing the lens and plaster-of-Paris reflector can be kept parallel 
to the sun’s rays. The heliotroper, standing behind the mirror 
and looking through the central hole towards the instrument 

















HELIOTROPES AND NIGHT-SIGNALS. 


573 


station, sees an imaginary sun produced by the reflection of 
the true sun from the plaster of Paris and focused by the lens 
on the surface of the glass. The mirror should then be slowly 
moved until this imaginary sun, moving with it, appears to 
rest on the object towards which the flash is to be cast. As 
both surfaces of the mirror are parallel, the true reflection of 
the sun from the surface of the mirror will also be cast on the 
object sighted. 

Various attempts have been made to design heliotropes 
which shall automatically follow the path of the sun in a 
manner similar to the clockwork mechanism employed in ob¬ 
servatories for following the movement of stars with large 
telescopes. None of these have proven satisfactory, how¬ 
ever, because of their complexity and weight. Other at¬ 
tempts at effecting a similar result have been made by using 
rectangular polished steel bars made to revolve about a hori¬ 
zontal axis, and the latter to revolve about a vertical axis 
through hand mechanism. An automatic motion for the 
same has been attempted by use of cup-shaped wind vanes, 
similar to those used in anemometers, whereby a many-sided 
heliotrope is made to revolve in all directions continuously 
by the wind. Such an apparatus flashes light in every direc¬ 
tion, but as yet such flashes have not been procured of suffi¬ 
cient duration and certainty to serve the purposes desired. 

The greatest objections to the use of heliotropes as signals 
are due to the uncertainty of the atmosphere and the diffi¬ 
culties of communication between observer and heliotroper. 
Where any attempt is made to observe on several helio- 
troped stations at one time, if the sun be occasionally or 
partly obscured by cloud, uncertainty arises on the part of 
the heliotroper as to whether the observer has measured all 
the angles required, and he may prematurely leave his station, 
thus causing considerable delay in conveying directions to 
him to return. To obviate this a brief code of heliograph 
signals should be arranged where much heliotroping is to be 


574 FIELD-WORK OF PRIMARY TRIANG ULA TION. 

done. The fewest possible sentences should be devised and 
practiced by heliotropers and observer. The method of con¬ 
veying such signals is by intermittent flashes and blank 
spaces, corresponding in general to the dots and dashes of the 
Morse telegraphic code. The interval between the period 
during which the heliotrope is permitted to shine and that 
during which its light is cut off by interposing the hand or 
some other object before the mirror to produce a blank, 
should be of not less than ten seconds’ duration nor as great 
as one minute. Thus a sentence may be conveyed by a flash 
of ten seconds followed by a blank of ten seconds, or another 
by a flash of ten seconds followed by a blank of thirty seconds 
and a flash of ten seconds. Any number of similar combina¬ 
tions may be prearranged. 

247. Night-signals.—Where the observation of angles is 
impeded during the daytime by dense smoke, the best results 
are procured by signalling with lights used at night. The 
moisture in the air at night carries the smoke down into the 
valleys, thus clearing the atmosphere between the higher 
summits from which observations are made. Several forms 
of night-signals have been employed with some success both 
in India and France, and experimentally by the U. S. Coast 
Survey. These lights are practically of three kinds only: 
(1) electric arc light, (2) magnesium tape, and (3) kerosene-oil 
lamp. All should be used with a parabolic reflector 12 or 
more inches in diameter, depending on the distance. 

The chief conditions in connection with a suitable night- 
light are that it should be (1) cheap, (2) capable of manipula¬ 
tion by persons of ordinary intelligence, (3) light enough to 
be easily transported to mountain-tops, and (4) simple of con¬ 
struction and adjustment. 

The form of light which fulfills these conditions best, ex¬ 
cepting that of cost, is magnesium tape. Experiments with 
this by the Coast Survey indicate that its cost is about $2.25 
per oz. of 40 yards length, and its consumption 12 to 18 


STATION- AND WITNESS-MARKS. 


575 


inches per minute if sufficient brilliancy and steadiness is 
maintained. This is at the rate of about 2 \ cents per minute, 
or $1.40 per hour if burned steadily. Accordingly, on the 
assumption of the average period for observing, which is 
about two hours, such night-signals will cost about $3 per 
night for each signal burned. Another form of night-signal, 
which is difficult to transport, but is perhaps even more satis¬ 
factory, is the ordinary kerosene-oil headlight of a locomotive. 

248. Station- and Witness-marks.—Primary triangula¬ 
tion stations should be so permanently marked as to render 
them possible of identification at any future time. This class 
of marking should include both surface and underground 
marks. The surface mark, being visible, is readily found, and 
in searching for a station its position can be verified by the 
discovery of the underground mark. Should the surface 
mark be disturbed, which is not unlikely, the witness-marks 
will indicate the positions of the underground mark, the dis¬ 
covery of which will again locate the station. 

Underground marks should be buried below frost and 
plow line, say at least three feet beneath the surface. Their 
chief characteristics should be: (1) indestructibility ; (2) peculi¬ 
arity of shape and appearance; (3) cheapness and lack of value 
as a protection against cupidity. Some of the following are 
excellent underground marks: a stoneware tile or tablet, 
dish or cone; a short chiseled block of granite, sandstone, or 
other stone not indigenous to the locality; a brick or block 
of hydraulic cement stamped with suitable lettering. 

Surface marks depend largely upon the nature of the 
material composing the ground surface. Where this is soil, 
they may consist (1) of iron posts sunk into the ground 
(Fig. 100) with a few inches projecting and bearing a suitable 
inscription on a metal cap; (2) of stone posts dressed to a 
square cross-section and appropriately marked ; or (3) where 
the surface is of rock a small eminence or a cross-mark should 
be chiseled on it and a copper bolt sunk therein. 


5 76 FIELD-WORK OF PRIMARY TRIANGULATION. 

Witness-marks are established by measurement and mag¬ 
netic bearings, which are recorded from the station-mark to 
projecting rocks, houses, trees, or to witness monuments 
which may be planted for the purpose. The record of a 
station should include descriptions of its summit, under¬ 
ground station-marks, and witness-marks, with a sketch of the 
whole. 


CHAPTER XXVI. 


MEASUREMENT OF ANGLES. 

249. Precautions in Measuring Horizontal Angles.— 

The following are some of the more important precautions to 
be observed in occupying a station and setting up the instru¬ 
ment preparatory to the measurement of horizontal angles, 
namely: 

1. Stability of support of instrument; 

2. Stability of foot-screws; 

3. Freedom of motion of alidade; 

4. Knowledge of signals; and 

5. Avoidance of gross errors in record. 

The instrument should have a stable support , which may 
be a stone pier, a wooden post, or a good tripod. If a port¬ 
able tripod is used, its legs should be set firmly in the ground 
and clamped tightly to the tripod head. On this the instru¬ 
ment should rest freely without being held by center clamp. 

The foot - screws of the instrument should be tightly 
clamped after it is leveled for work. Looseness of the foot- 
screws and tripod is a common source of error. 

The alidade , or part of the instrument carrying the tele¬ 
scope, circle, and verniers or microscopes, should move freely 
on the vertical axis. Clamps should likewise move freely 
when loosened. Whenever either of these moves tightly, the 
instrument needs cleaning, oiling, or adjusting. 

The observer should always have a definite preliminary 
knoivledge of the signals or objects observed. The lack of it 

577 




578 


MEASUREMENT OF ANGLES. 


may lead to serious error and entail cost much in excess of 
that involved in procuring such knowledge. 

Great care should be taken to insure correctness i?i the 
record of degrees and minutes of an observed angle. The 
removal of an ambiguity in them is sometimes a troublesome 
and expensive task. 

250. Observer’s Errors and their Correction. —The 

general directions and precautions given in Articles 250 to 
252 regarding instrument errors and the measurement of 
angles in a system of primary triangulation were prepared by 
Prof. R. S. Woodward for the guidance of the observers of 
the U. S. Geological Survey. They are supplemented by 
memoranda from the Proceedings of the Geodetic Conference 
held at the U. S. Coast Survey office in Washington in 1893, 
and from the experience of the author. 

The errors to which measured angles are subject may be 
divided into two classes, viz. : 

1. Those dependent on the instrument used, or instru¬ 
mental errors; and 

2. Those arising from all other sources, which may be 
called observer’s or extra-instrumental errors. 

Extra-instrumental errors may be divided into four 
classes, namely: 

1. Errors of observation ; 

2. Errors from twist of tripod or other support; 

3. Errors from centering; and 

4. Errors from unsteadiness of the atmosphere. 

Barring blunders or mistakes, the errors of observation are 

in general relatively small or unimportant. With observers 
practiced in measuring angles such errors are the least for¬ 
midable of all the unavoidable errors, and the methods 
devised for their elimination result in practical perfection 
The recognition of this fact is very important, for observers 
are prone to attribute unexpected discrepancies to bad ob¬ 
servation rather than to their much more probable causes. 


OBSERVER'S ERRORS AND THEIR CORRECTION. 579 

After learning how to make good observations the observer 
should place the utmost confidence in them, and never yield 
to the temptation of changing them because they disagree 
with*some preceding observations. Such discrepancies are 
in general an indication of good rather than of poor observa¬ 
tions. 

Stations or tripods which have been unequally heated by 
the sun or other source of heat usually twist more or less iiv 
azimuth. The rate of this twist is often as great as a second 
of arc per minute of time, and it is generally quite uniform 
for intervals of ten to twenty minutes. The effect of twist 
is to make measured angles too great or too small, according* 
as they are observed by turning the microscopes in the direc¬ 
tion of increasing graduation or in the opposite directions 
This effect is well eliminated, in general, in the mean of two 
measures, one made by turning the alidade in the direction 
of increasing graduation, followed immediately by turning 
the alidade in the opposite direction. Such means are called 
combined measures or combined results, and all results used 
should be of this kind. As the uniformity in rate of twist 
cannot be depended on for any considerable interval, the 
more rapidly the observations of an angle can be made the 
more complete will be the elimination of the twist. The 
observer should not wait more than two or three minutes 
after pointing on one signal before pointing on the next. If 
for any reason it should be necessary to wait longer than, 
such short interval, it will be best to make a new reading on: 
the first signal. 

The precision of centering an instrument or signal over the 
station or geodetic point increases in importance inversely 
as the length of the triangulation sides. Thus if it is desired 
to exclude errors from this source as small as a second, one 
must know the position of the instrument within one-third of 
an inch for lines a mile long, or within six inches for lines 
twenty miles long. The following easily remembered rela- 


580 measurement of angles. 

tions will serve as a guide to the required precision in 
any case: 

i second is equivalent to 0.3 inch at the distance of 1 mile; 


J it < < 

< i 

“ 3.0 inches “ “ 

< < 

“10 miles; 

J < < it 

11 

“ 6.0 “ “ “ 

< < 

“20 “ ; 

1 minute “ 

< < 

“ 1.5 feet ** “ 

i ( 

“ 1 mile. 


The notes should always state explicitly the relative posi¬ 
tions of instrument and signal, and give their coordinates 
(preferably polar coordinates) if they are not centered. 

Objects seen through the atmosphere appear unsteady, 
and sometimes this boiling of the atmosphere is so great as 
to render the identity of objects doubtful. This is usually 
greatest during the middle of the day, and generally subsides 
or ceases for a considerable period between 2 P.M. and sun¬ 
down. There is frequently also a short interval of quietude 
about sunrise, and on cloudy days many consecutive hours of 
steady atmosphere may occur. For the best work, observa¬ 
tions should be made only when the air causes small or 
imperceptible displacements of signals. In applying this 
rule, however, the observer must use his discretion. Errors 
of pointing increase rapidly with increase of unsteadiness, but 
it will frequently happen that time may be saved by counter¬ 
balancing errors from this source by making a greater number 
of observations. Thus, if signals are fairly steady, it may 
be economical to make double the number of observations 
rather than wait for better conditions. 

251. Instrumental Errors and their Correction.—The 
best instruments are more or less defective, and all adjust¬ 
ments on which precision depends are liable to derangement. 
Hence results the general practice of arranging observations 
in such a manner that the errors due to instrumental defects 
will be eliminated. The principal errors of this kind and 
the methods of avoiding their effects are: 

1. Periodic errors ; 

2. Accidental errors; 


INSTRUMENTAL ERRORS AND THEIR CORRECTION. 581 

K!i a ■ * 'liiado j': 13 rdo 

3. Collimation errors; , u 

4. Errors due to inequality of pivots; 

5. Errors due to inequality in height of wyes; 

6. Errors due to inclination of telescope axis; 

7. Errors due to parallax of cross-hairs; 

8. Errors of run of micrometer-screw; and 

9. Errors from loose tangent or micrometer screws. 
Measurements made with a graduated circle are subject to 

certain systematic errors commonly called periodic. Certain 
of these errors are always eliminated in the mean or sum 
of the readings of the equidistant verniers or microscopes, 
and both or all of these should be read with equal care in 
precise work. Certain other errors of this class are not elimi¬ 
nated in the mean of the microscope readings, and only 
these need consideration. Their effect on the mean of all the 

, k 

measures of an angle may be rendered insignificant by making 
the same number of individual measures with the circle in 
each of n equidistant positions separated by an interval equal * 

to where m is the number of equidistant verniers or 


nm 

microscopes 


Thus, if m — 2, the circle should be shifted 

, , 180 0 f . 

after each measure by an amount equal to ——, which, for 


example, is 45 0 for n = 4 and 30° for n — 6. The degree of 
approximation of this elimination increases rapidly with n. 
(Art. 252.) Other things being equal, therefore, the measures 
of such special angles should show less range than the measures 
of other angles. 

Besides the instrumental errors of the periodic class, there 
are also accidental errors of graduation. These cu*e in gen¬ 
eral small, however, in the best modern circles, and their 
effect is sufficiently eliminated by shifting the circle in the 
manner explained above for periodic errors. 

The effect of an error of collimation on the circle reading 
for any direction varies as the secant of the altitude of the 




582 


MEASUREMENT OF ANGLES. 


object observed. The effect on an angle between two objects 
varies as the difference between the secants of their altitudes. 
This effect is eliminated either by reversing the telescope in 
its wyes, or by transiting it without changing the pivots in 
the wyes, the same number of measures being obtained in each 
of the two positions of the telescope. The latter method is 
the better, especially in determining azimuth, since it elimi¬ 
nates at the same time errors due to inequality of pivots and 
inequality in height of wyes. 

The effect of errors due to inclination of telescope axis on 
the circle reading for any direction varies as the tangent 
of the altitude of the object observed. If the inclination 
is small, as it may always be by proper adjustment, its effect 
will be negligible in most cases. But if the objects differ 
much in altitude, as in azimuth work, the inclination of the 
axis must be carefully measured with the striding-level, so 
that the proper correction can be applied. The following 
formula includes the corrections to the circle reading on any 
object for collimation and inclination of telescope axis: 

Cor. = c sec li + b tan h\ . . . . (51) 

in which c = collimation in seconds of arc; 

b — inclination of axis in seconds of arc; 
h = altitude of object observed. 

Parallax of cross-hairs occurs when they are not in the 
common focal plane with the eyepiece and objective. It is 
detected by moving the eye to and fro sidewise while looking 
at the wires and image of the object observed. If the wires 
appear to move in the least, an adjustment is necessary. The 
eyepiece should always be first adjusted to give distinct vision 
of the cross-hairs. This adjustment is entirely independent 
of all others, and requires only that light enough to illuminate 
the wires enter the telescope or microscope tube. It is de¬ 
pendent on the eye, and is in general different for different 
persons. Hence bad adjustment of the eyepiece cannot be 


INSTRUMENTAL ERRORS AND THEIR CORRECTION. 583 

corrected by moving the cross-hairs with reference to the ob¬ 
jective. Having adjusted the eyepiece, the image of the 
object observed may be brought into the plane of the cross¬ 
hairs by means of the rack-and-pinion movement of the tele¬ 
scope. A few trials will make the parallax disappear. 

When circles are read by micrometer microscopes it is 
customary to have them so adjusted that an even number of 
revolutions of the screw will carry the wires over the image of 
a graduation space. If the adjustment is not perfect, an error 
of run will be introduced. This may in all cases be made 
small or negligible, since by means of the independent move¬ 
ments of the whole microscope and the objective with respect 
to the circle the image may be given any required size. In 
making this adjustment some standard space, or space whose 
error is known, should be used. At least once at each station 
where angles are read observations should be made for run of 
micrometers. 

READINGS FOR RUN OF MICROSCOPES ON SPACE 359’ 50' 

TO 360°. 

(Ideal case showing microscopes in need of adjustment.) 


A-, ,-B 


359" 5o' 

360 ° 

359° 50' 

360 ° 

4.O 

3 -i 

i -7 

0.2 

4.0 

2.2 

2.1 

1.1 

3.9 

2.4 

2.0 

0.7 

3-3 

2.6 

i -7 

0.0 

4.1 

2.7 

2.1 

0.1 

Means.3.86 

2.60 

1.92 

0.42 

Difference.- 

-1.26 


-^1.50 

Error of space. . . . - 

-0.37 known 


— 0.37 known 

Error of run.- 

-1.63 for 5 revs. 

— 1.87 for 5 revs. 

Hence readings 

of microscope 

A should 

be diminished by 


0.33 div. per revolution, and those of B by 0.37 div. per 
revolution, which is one-fifth of the error of run in each case. 













584 


MEASUREMENT OF ANGLES. 


Errors from loose tangent or micrometer screws are due to 
their moving too freely or loosely. In making a pointing 
with the telescope the tangent screw should always move 
against or push the opposing spring. Likewise, bisections 
with the micrometer wires must always be made by making 
the screw pull the micrometer frame against the opposing 
spring or springs. 

252. Methods of Measuring Horizontal Angles. —Two 

general methods are employed for reading parts of the angle 
less than the smallest space graduated on the horizontal limb, 
namely: 

1. By means of verniers; and 

2. By micrometer microscopes. 

Where it is unnecessary to read angles to lesser amounts 
than 10 seconds of arc verniers may be successfully employed. 
If greater accuracy is to be attempted by reading to smaller 
fractions of the arc, micrometer microscopes must be em¬ 
ployed. Primary angles are read with verniers by the method 
of repetition, and with micrometer microscopes by the method 
of directions. 

Vernier or repeating theodolites are not used now to any 
extent on primary work of high order. In order that the 
best results may be had from such an instrument it must 
have a very large circle, as from 16 to 20 inches diameter, 
and be proportionately heavy and cumbersome. Such instru¬ 
ments are now generally employed only on secondary tri¬ 
angulation, where a circle not greater than six or seven inches 
in diameter will give satisfactory results. 

The method of reading angles with the repeating instru¬ 
ment consists in pointing at the first station, n , and with the 
lower circle clamped revolving the graduated limb and point¬ 
ing at the second station, 0 (Fig. 175). Then with the upper 
circle clamped the instrument is revolved on its lower circle 
in the reverse direction so as to point back again at n, and 
the operation is repeated. By this means the direct reading 


METHODS OF MEASURING HORIZONTAL ANGLES. 585 

of the angle between the two stations is increased or added 
on the vernier in proportion to the number of repetitions of 
the angle made. Thus if six such angles are read, the single 
angle will be the total recorded on the circle divided by 6, 
and the result will be a reading possibly | smaller than the 
amount by which a single angle could be read on the vernier. 
In the use of such an instrument each set of repetitions con¬ 
sists of a fixed number of measures of the angle, say three, 
followed by an equal number of measures with the telescope 
reversed. Two sets of six repetitions, as 3 direct -f- 3 reversed, 
are preferable to one set of twelve repetitions, as 6 direct -j- 6 
reversed, because something may occur to interrupt the 
observations during the longer time. In like manner the 
various angles between the adjacent stations observed are each 
separately read. 

The best results procurable in the measurement of hori¬ 
zontal angles are obtained with direction instruments. Such 
instruments, with circles as small as 8 inches, will give more 
accurate readings of the angle than a corresponding repeating 
instrument of 16 to 18 inches circle. In observing with a 
direction instrument the more usual method is to divide the 
circle into a number of equal parts known as positions. This 
number should be such that no microscope may fall upon the 
same graduation in pointing upon the same object in different 
positions or after reversal of the telescope. Having estab¬ 
lished the initial direction, one or more series are observed in 
each position, each consisting of the pointing and reading 
upon each of the signals in order and reading of the gradua¬ 
tion of the circle. Then the telescope is reversed , the alidade 
turned 180° in azimuth, and another pointing and reading 
made upon the various signals in order. The number of the 
various positions depends upon (1) the accuracy of the gradu¬ 
ation, and (2) upon the degree of refinement desired. For 
geodetic work of a high order from twenty-four to thirty 
positions or series should be observed. For primary triangu- 


586 


MEASUREMENT OL ANGLES. 


lation of a sufficiently high order for map-making purposes, 
however, six to eight positions are sufficient. 

Angles may be read with a direction instrument by two 
general methods, namely: 

1. Method of independent measures; and 

2. Method of measurement by series. 

The best results are obtained by measuring the angles 
separately and independently. Thus if the signals in sight 
around the horizon are in order n, o, p, etc. (Fig. 175), the 
angles n to 0 , 0 to /, etc., are by this method observed 
separately, and whenever there is sufficient time at the 
disposal of the observer this method should be followed. 

In order to secure the elimination of the errors of obser¬ 
vation (Arts. 250 and 251) the following programmes should 
be strictly adhered to. 

When direction instruments are used the following is the 
programme for independent measurement of angles : 

Pointing on n and readings of both micrometers. 

< t < < Q i ( << < < (l < ( 

Transit telescope and turn alidade 180°. 

Pointing on 0 and readings of both micrometers. 

“ “ n “ “ “ “ “ 

180 0 

Shift circle by - ^v — and proceed as before until N such' 

sets of measures have been obtained. 

Then measure the angles 0 to p, p to <7, etc., including the 
angle necessary to close the horizon, in the same manner. A 
form for record and computation of the results is given in 
Article 253. 

When repeating instruments are used the same programme 
will be followed, except that there should be five pointings 
instead of one each on n and 0, the circle being read for the 
first pointing on n and the fifth on o, and again for the sixth, 
pointing on 0 and the tenth on n. 

The importance of having the measures of a set follow irk 



METHODS OF MEASURING HORIZONTAL ANGLES. 5 87 


quick succession must be constantly borne in mind. Under 
ordinarily favorable conditions an observer can make a point¬ 
ing and read the microscopes once a minute, and a set of five 
repetitions should be made in five minutes or less. 

When several stations or signals are visible and a direction 
instrument is used, time may be saved without material loss 
of precision in the angles by observing on all the signals suc¬ 
cessively according to the following programme for measure¬ 
ment by series , the signals being supposed in the order n, 0 , p , 
etc., as above : 

Pointing on n with micrometer readings. 

i l i t Q t l It ( i 

i < iij.it u l i 


Pointing on n with micrometer readings. 
Transit telescope and turn alidade 180 0 . 
Pointing on n with micrometer readings. 

< < i i y ( t l ‘ ‘ < 

i ( l l n i ( l l it 


Pointing on n with micrometer readings. 

18o° 

Shift circle by - and proceed as before until N such 
sets have been obtained. 

The angles n to 0, 0 to p, etc., read in this way may be 
computed as in the first method, always combining the meas¬ 
ure n to 0 with the immediately succeeding measure 0 to n to 
eliminate twist. There is a theoretical objection to this 
process of deriving the angles founded on the fact that they 
are not independent, but in secondary work this objection 
may be ignored as of little weight. 

In observing horizontal angles the number of sets of meas¬ 
ures of any angle is dependent upon the character of instru¬ 
ment and the precision desired. For the primary triangula- 





588 


MEASUREMENT OF ANGLES. 


tion of the U. S. Geological Survey with 8-inch direction 
theodolite read by micrometer microscopes, four sets of 
measures on as many different parts of the circle will be 
required. For repeating theodolites six sets of measures 
will be required, all made according to the programmes given 
above. Only under specially unfavorable conditions will 
it be necessary to increase the number of sets of measures. 
Care should always be taken to shift the circle so as to elimi¬ 
nate periodic errors. 

When there is ample time at the disposal of the observer, 
or need for additional measures, the work may be strength¬ 
ened by measuring sum-angles. This is done in such manner 
as to introduce additional conditions which will thus strengthen 
the least-square adjustment. Thus, after reading the separate 
angles n to o, o to />, p to q , etc., the intermediate pointings 
may be skipped by reading from n to p, p to r , etc., and the 
conditions are introduced that n to o -j- o to p — n to /, 
and o to / -f- / to q = o to q. 

The practice of starting the measurement of an angle or 
series of angles with the microscopes reading o° and i8o°, 90° 
and 270°, etc., must be avoided; otherwise the errors of these 
particular divisions will affect many angles. In shifting the 
circle it is neither necessary nor desirable to have the new 


position differ from the preceding 


one by exactly 


180 0 

~aT' 


A 


difference of half a degree either way is unimportant as re¬ 
spects periodic errors, and it is advantageous to have the 
minutes and seconds differ for the different settings. 

253. Record of Triangulation Observations. —In record¬ 
ing the angles read at any primary station with the theodolite, 
the first page of the notes should give a concise description 
of the station, how it is reached, character of station-mark, 
description of witness-points, and a topographic sketch of 
station and surroundings. There should also be, in case of 
the necessity of reduction to center (Art. 267), (1) a diagram 



RECORD OF TRIANGULATION OBSERVATIONS. 589 

showing the relation of the signal to the position of the 
instrument, (2) the distance between the two, and (3) the 
angle read at the instrument position between one or more 
of the observed stations and the signal, that full data for 
reduction may be available. (Fig. 174.) Another diagram 
should show the directions to the various stations observed, 
and the arrangement or groupings of the angles. (Fig. 175.) 
The date and time of observation should be noted at inter¬ 
vals, to show that the instrument has not stood too long 
between pointings. 

The following is an example of the record made of point¬ 
ings from the triangulation station “Township ” occupied in 
Kansas by the U. S. Geological Survey in 1889. This is the 
record of one pair of pointings only, that determining the 
angle observed between the stations Newt and Walton. 

RECORD OF MEASUREMENT OF HORIZONTAL ANGLE. 

H. L. Baldwin, Observer. 

(Station: Township corner, Kansas. July 1, 1889. Fauth 8-inch theodolite No. 362; one divi¬ 
sion of micrometer head = 2 seconds.) 


Station. 

Micr. A. 

Micr. 

B. 

Mean Reading’. 

Angle. 

Mean. 



Telescope direct. 








O 

' Div . 

O / 

Div . 

O 

/ 

/; 

O / // 

// 

Walton . 

93 

12 11.3 

273 12 

09.9 

93 

12 

21.2 

36 29 03.9 


Newt. 

129 

41 11.9 

3°9 4 i 

13.2 

129 

41 

25-1 


° 5-9 

Newt. 

129 

41 15.6 

3°9 4 1 

12 . I 

129 

4 » 

27.7 

08.0 


Walton. ... 

93 

12 10.6 

273 12 

09.1 

93 

12 

19.7 





Telescope reversed. 







Walton . 

138 

27 03.2 

318 26 

28.0 

138 

27 

01.2 



Newt. 

174 

56 02.8 

354 55 

28.9 

174 

56 

01.7 

00.5 


Newt. 

T 74 

56 06.2 

354 55 

29.5 

*74 

56 

05-7 


01.8 

Walton. 

>38 

27 05.2 

318 26 

27.4 

138 

27 

02.6 

03.1 




Telescope reversed. 







Walton. 

183 

07 03.0 

3 06 

27.2 

183 

07 

00.2 



Newt. . 

219 

36 05.0 

39 35 

29.8 

219 

3 6 

04.8 

04.6 


Newt . 

219 

36 08.1 

39 35 

29-5 

219 

36 

07.6 


03.9 

Walton. 

183 

07 06.4 

3 06 

28.1 

183 

07 

04*5 

03.1 




Telescope direct. 







Walton.. . 

228 

24 28 1 

48 24 

22.6 

228 

24 

50.7 



Newt. 

264 

53 27-4 

84 53 

26.1 

264 

53 

53-5 

02.8 


Newt. 

264 

54 01 - 1 

84 53 

26.1 

264 

53 

57-2 


04-3 

Walton. 

228 

24 29.3 

48 24 

22.1 

228 

24 

5 1 *4 

05.8 



4 | 1 5 // »9 

Mean of four combined measures. 36° 29' 03".98 



































590 


MEASUREMENT OF ANGLES. 


254. Instructions for Field-work of Primary Triangu¬ 
lation. —The following instructions are those governing the 
field-work of primary triangulation in the U. S. Geological 
Survey. 

1. Signals should be of sawed lumber whenever it can be 
obtained, and great care must be taken to secure perfect 
centering of instrument and target over station-mark. 

2. All stations should be selected with a view to their 
adaptability to topographic expansion, and when the exact 
location of a station is decided upon one of the standard iron 
posts, copper plugs, or bronze tablets must be set as a per¬ 
manent mark. In light soil a bottle or similar object must 
be left as a subsurface mark. These marks should be at 
exact center of station, and in addition there should be left 
one or more reference marks. At base-line stations there 
should be left at least two reference marks. 

3. Whenever practicable, set the theodolite over center 
of station while reading angles, to obviate reduction to 
center. 

4. The theodolite when in use must be sheltered from the 
sun and wind. When setting the theodolite tripod, leave 
the tripod-head thumb-screws loose until the legs are firmly 
placed. 

5. Never, under any circumstances, attempt to place the 
circle so that when pointing at any particular station the 
micrometers will be set to even degrees. 

6. Use book No. 9-912 for all field records, and do not 
crowd notes. Have notes plainly written with No. 4 pencil 
or with ink, and never erase, but draw a single line through 
erroneous records. 

7. On page immediately preceding record of angles, write 
a minute and complete description of the station occupied, 
giving nearest trails or roads, camping-places, station-marks, 
etc., as well as ownership of land when possible. Write this 
description before leaving the station. In addition plat a 


INSTRUCTIONS FOR PRIMARY TRIANG ULA TORS. 591 

rough diagram of pointings, showing also plan of eccentric 
location of instrument, if there be such. 

8. Before observations are commenced at a station, test 
all adjustments of theodolite, and correct such as are found in 
error, paying special attention to micrometers to avoid the 
errors of run. 

9. For micrometer theodolites, angles must be measured 
either by the method of circle readings (directions) or by single 
angles, and in either case each set of angles must be kept on 
a single page of note-book. If the method of directions be 
adopted, each complete set must consist of pointings with 
telescope direct, and.reverse pointings with telescope inverted, 
always closing horizon. 

10. No angle should be considered finally determined that 
has not been measured on at least four different parts of the 
circle. 

11. The error of closure of any triangle in primary schemes 
should not exceed 5". 

12. Opposite each angle recorded give any necessary 
information in regard to visibility of signals or atmospheric 
conditions. 

13. Do not trust to memory for notes. Make all notes as 
complete as though it were expected another person would 
compute them. 

14. Magnetic declination must be determined at each 
azimuth station and at each county seat. 

15. Observations for azimuth on Polaris before and after 
elongation must be made on two nights from at least one 
station in each square degree, to consist of not less than 6 
angles between mark and star with telescope direct and 
reversed. See Monograph above referred to for form of 
record. Great care must be taken in adjusting and leveling 
the horizontal axis of theodolite. Watch error must be deter¬ 
mined by telegraphic comparison of time or by astronomic 
observations. 


592 


MEASUREMENT OF ANGLES. 


16. Two marks of dressed stone or masonry, about 500 
feet apart on a true north-and-south line, must be established 
at each county seat, the center of each to be the cross-mark 
on one of the standard bronze tablets. 

17. Angles at each station must be reduced to center of 
permanent mark in order to test triangle closures. Arbitrary 
adjustments and preliminary computations should be made in 
the field. All computations except distances and coordinates 
must be in book No. 9-889. 

18. Keep a careful plot of the work on a scale of 10 miles 
to an inch, and each month send a copy with monthly report, 
indicating angles measured by the usual signs. 

19. On fly-leaf of each note-book write an index of con¬ 
tents of book, and state make and number of theodolite used. 

20. The observer should always endeavor to locate promi¬ 
nent points that may be of use to the topographer, or that 
may be used for future stations. 

21. Especial attention must be paid to the location of 
county court-houses, section and county corners, and State¬ 
line marks. 

22. Useful locations can often be made by the “three- 
point method,” the theodolite being set up for the purpose 
while going to or from stations. 

23. Keep in view the fact that station names are to be 
published, and select such as have local significance. 

255. Primary Triangulation—Cost, Speed, and Accu¬ 
racy. —Triangulation of the highest geodetic precision, as 
executed by the U. S. Coast and Geodetic Survey, costs at 
the average rate of $1500 per station occupied and from $10 
to $30 per square mile, according to the character of the 
topography; the daily cost of a party of from five to fifteen 
individuals averaging $65 The speed of the work, or, in 
other words, the length of time which is required to occupy a 
station, is indicated by the rate of J station per month. In 
this work the average closure error of a triangle is o".y, the 


COST, SPEED, AND ACCURACY. 


593 


probable error of an extended triangulation being y ¥¥ V(nr* 
Or, stated otherwise, a line io miles in length would have a 
probable error of 0.35 ft. 

In the primary triangulation executed by the U. S. Geo¬ 
logical Survey, not for geodetic purposes, but with sufficient 
accuracy to safely control topographic maps, the average cost 
per station is $170, and the cost per square mile controlled 
about 90 cents. The cost per day for working-parties of 
from two to five members has averaged $18. The speed has 
been at the rate of six stations per month. The accuracy 
is shown by closure errors averaging 3T0. The probable 
error of this triangulation has averaged - 4 - 0 - ^- 0 ¥ , which may be 
otherwise expressed as 1.32 feet in a line 10 miles in length. 


CHAPTER XXVII. 


SOLUTION OF TRIANGLES. 


256. Trigonometric Functions.—Let a — angle GAB 
= arc GB, and let radius AB — AG = 1 ; then 



Fig. 168. —Trigonometric 
Functions. 


sin a = CB ; 
cos a — AC; 
tan a — HG 
cot an a = DE; 

sec a — AH 
cosec a — AE ; 
versin a — BE; 
coversin a = BI; 
chord a — GB. 


257. Fundamental Formulas for Trigonometric Func¬ 
tions.—The fundamental formulas are: 


sin* a -f- cos 2 a == 1 ; 


tan a cot a = 1 ; 


cos a sec — \ ; 

sin a 

tan a = 


cos a 


sin a cosec a 
cot a 


1 ; 

cos oe 


sin a 


I -1- tan 3 a = —— = sec 3 a; 1 4- cot 3 a = . 2 
1 cos a sin a 

versed sin a = 1 — cos a. 


= cosec 3 a ; 


258. Formulas for Solution of Right-angled Tri¬ 
angles. —In the right-angled triangle, Fig. 168, 


594 










SOLUTION OF RIGHT-ANGLED TRIANGLES . 595 


Let a — altitude, 
b — base, and 
c — hypothenuse ; and let 

a, ( 3 , and y = the angles opposite a } b , and c, respectively; 

also let 

A = area of triangle, and 
R == radius of circumscribed circle. 


For a right-angled triangle y = 90° ; the fundamental 
values of a , b , and A are then 


a — c sin a = c cos — b tan a — b cotan yd; 
b = c sin = c cos a — a tan fi = a cotan a ; and 
A = cotan a — tan a = Jr 2 sin 2«. 


Fig. 169 furnishes a method of graphically stating the 
formulas relating to the solution of right-angled triangles. 


Fig. 



I 6 9 ._Graphic Statement of Formulas for Solution of 

Right-angled Triangles. 


Let P — perpendicular in a right-angled triangle, the 
angle between the base of which, B , and the hypothenuse, H , 
is denoted by a. 

Then the diagram is applied by the use of the following 
rules, the order of sequence being to follow either the names 




596 


SOLUTION OF TRIANGLES. 


written around the circumference of the circle or by following 
the names along the intersecting lines in the order written; 
thus: 

i. Any trigonometric function or part equals the adjacent 
part divided by the following part. Example: 

cos a 

sin a = - ; 

cot a 

also, sin a = —, 


’ cos a 

2. Any part equals the product of the adjacent parts. 
Example: 

a — c sin a — b tan a ; cosin a = sin a cotan a. 

3. Each part equals the reciprocal of the opposite part 
Example: 

* 1 1 

tan = —--; sec a = - : -. 

cotan a cosin a 

4. The product of opposite parts equals 1. Example: 

tan a cotan a = 1. 

259. Solution of Plane Triangles. —In the solution of 
geodetic triangulation there arise a few simple problems 
which involve the solution of triangles in accordance with 
the principles of trigonometry. These occur when one or 
more angles or sides have been measured in the field and the 
dimensions of the remaining parts are desired. In the fol¬ 
lowing articles are illustrated by practical examples those 
problems most likely to arise in actual practice. 

Table XXXIII, from Smithsonian Tables, gives all the 
more important formulas for finding unknown parts of a 
triangle with three parts given. 






SOLUTION OF PLANE TRIANGLES. 


597 


Table XXXIII. 

SOLUTION OF OBLIQUE PLANE TRIANGLES. 


Given. 

Sought. 

Formula. 

a, b, c 

a 

sin \a = |/ ( S ~ /Xf - fl, s = \(a + b -{- c), 

cos {a = a \ 

tan ia = i/ (s ~ ^ ~ C \ 

2 Y s(s — a) 


A 

A = V s(s — a)(s — — c). 

a, b y (X 

ft 

sin ft = b sin a/a. 

When a > b, ft < go° and but one value results. When 



b > a, ft has two values. 


V 

y — i8o° — (a + ft). 


c 

c = a sin y / sin a. 


A 

A = \ab sin y. 

a, a, ft 

b 

b — a sin /5/sin a. 


r 

y = i8o° — (« -f- ft). 


c 

c — a sin y / sin a — a sin fa: -f- ft)/ sin a. 


A 

A = \ab sin y — |a 2 sin ft sin y / sin a. 

a, b, y 

a 

a sin y 

tan a = . 

b — a cos y 


a, ft 

*(« + ft) = 90° - hr, 



tan \{oc - ft) - a ^cot \y. 


c 

c = (a 1 + b* ~ 2ab cos r)l> 



= {(a + by — 4 “b cos 2 \y }*, 

= | {a — b) 2 -f- \ab sin" $y\^, 

= (a — b)/ cos (p, where tan 0 = 2 Vab sin \y/{a — b), 



= a sin y/ sin a. 


A 

A = \ab sin y. 





























598 


SOLUTION OF TRIANGLES . 


260. Given Two Sides and Included Angle, to Solve the 
Triangle. 


c 

py 

/ tan 

a' = 

log [j 

J (less from greater). . 

(52) 


tan 

(«'- 

- 45°) = 

= tan 8 (less from greater). 

(53) 

/^ 
ry 

tan 

a X 

tan i(a 

+ ft) = tan i(a — /?). 

(54) 



* + p) 

2~(^ — — (X. 

(55) 

Fig. 170. 


4(' 

<x + fi) 

— i( a - P) — P- • • 

(56) 


Knowing a and /?, compute remaining parts. 

Check is that the sum of the angles or a -)- fi -|- y = 180°. 
For convenience always call greater side a\ then as greater 
side is always opposite greater angle, ol is opposite a. 


Example. 


Let log a = 4.1361976, 
log b — 4.1726495 

tan a ' = 0.0364519 
a ' = 47 0 24' o6 ,/ . 1 
45° 

<5=2° 24' 06". 1 
180 0 

Let y = 51° 43 ' 27^.3 

2)128° 16' 32'.7 


log tan 6 = 8.6226496 
tan \{a + fi) = 0.3144757 

tan U a ~ fi ) = 8.9371253 

“ fi ) — 4° 56' 42".06 

!(a-f 0 ) = 64° 08' 16".35 
— fi ) — 4° 56' 42".06 

(5 = 59° 11' 34".29 

Check . 

a = 69° 04' 58 '.4i 
/i = 59° 11' 34"-29 
y - 51° 43' 27".30 


+ /?) = 64° 08' 16 '.35 

+ U a ~ ft ) = 4° 56' 42".o6 


a = 69° 04' 58".41 180° 00' 00".00 

261. Given Certain Functions of a Triangle, to Find 
Remainder. 


The sides of a triangle are proportional to the sines of 
their opposite angles, hence 

b sin a 

sin ^ • ••••• (57) 


a 



* 

















GIVEN THREE SIDES OF A TRIANGLE. 


599 


Example. —Let a = 12.92 miles; £=153 ft. 7 in* ; 
a — 130° 58' 18 ".3. 

Almost all field measures made in the United States are in 
miles, feet, etc., while all geodetic tables are prepared on the 
metric system ; hence the former must be reduced to the latter 
for computation. Reducing miles and feet to the same unit, 
meters, and finding the corresponding logarithms, we have 

log a — 4.31790 
a. c. log a — 5.682 10 
log b — 1.67035 
log sin a — 9.87796 

log sin (3 — 7.23041 

f 3 = oo° 05' 50''.62 


262. Given Three Sides of a Triangle, to Find the 
Angles. 

s = one-half the sum of the three sides. For con¬ 


venience designate 



( f ~ ^ - 6 * s ~ by H, then 


and 


tan ia = 
tan i /3 = 
tan \y — 


// 


j — a 
H 

s - V 
H 

s — c 


c — 

4.1908 



b = 

40.8954 



a = 

43.7566 



2 S = 

88.8428 



S = 

44.4214 

log 5 - 

1.6475922 

■ a — 

0.6648 

log = 

9.8226910 

■ b = 

3.5620 

< i _ 

O.5472823 

■ C —- 

40.2306 

( i _ 

I.6045564 


Sum of logs -|- 1.9745297 
log j = — 1.6475922 

H 3 = 

H = 


a 


y 


0.3269375 
O.1634688 Fig. 172. 











6 oo 


SOLUTION OF TRIANGLES. 


0.1634688 = // 

9.8226910 = log s — a 

0.3407778 = log tan \a = 65 0 28' 27"; . *. a = 130° 56' 54" 

0.1634688 (If) log 
.5472823 (s - b) log 

9.6161865 = log tan yS = 22 0 27' 06"; . *. (5 — 44° 54' 12" 

0.1634688 = log H 
1.6045564 = log j — r 

8.5589124 = log tan \y \y = 2° 4' 27 y = 4° S' 54" 

Check 180 0 00' 00" 


A. 


263. Three-point Problem. —The object sought in the 

solution of this problem is the determi¬ 
nation of the unknown position of an 
occupied station P, when the positions 
of three other stations, A, B, and C, are 
known. (See Graphic Solution, Art. 

75-) 

The problem is indeterminate when 
P is on the circumference of a circle 
passing through A, B , and C. This is 
known by the sum of the angles p -f- 
p' + c, being equal to 180°, and also 

P t ^ 

^ _ by the radius of the circumference pass- 

Fig. 173. —Three-point j r 

Problem. ing PAC , being equal to that for PBC. 



/ a sin p' \ 

cot X = cot R = l -r—z - 7 -• • (61) 

'0 sin p cos R * K ' 

in which 

1 . R = 360° — p — p' — c ox R — x — y ox R — x = y. 

If p -f- = c or nearly, the solution is impossible. 

, a sin p' . 

2. loc: ;—:- „ = log of a number taking the sign of 

b sin p cos R * b * 

cos R. 

3. Add algebraically + 1 to the above number. 

4. Take out the log of this number, annexing the proper 

sign. 










THREE-POINT PROBLEM. 


601 


5. Then add this log to log cot R } remembering that this is 

in effect multiplying one by the other, and the rule of 
the signs must be attended to; this gives the log of the 
cot of 4r. Then R — x — y. 

6 . UR < 90°, cos R is -f- and cot R is 

“ R < 270° and R > 180°, cos R is —, cot R is 

“ R < 180° and R > 90°, cos R is —, cot R is —. 

“ R < 360° and R > 270°, cos R is -f-> cot R is —. 

7. p' is at opposite side of quadrilateral to a, and p to b. 

8 . The angle c is always the interior angle of the quadri¬ 

lateral PBCA , and C is the middle point as seen from P. 

Example.—T he following quantities are known from observation or 
computation, since the positions of B, C, and A are known, namely : 


p = 20 0 05' 53"; 


a = 6672.47 ft.; 
p ' = 35° 06' 08"; 
b = 12481.66 ft.; 


c — 152 0 23' 22". 


Then 


R — 152 0 24' 37" = 360° — p — p * — c . 


log a — 3.8242868 
sin p ' — 9.759 6 958 
a.c. log b = 5.9037277 
a.c. sin p — 0.4639117 
a.c. cos R — 0.0524258 — 

. (cos R gives sign) 0.0040478 — = the number — 1.009364 



a sin p' 


-f-1.000000 R — 152 0 24' 37'' 
— 0.009364 x — 88° 58' 24" 


y = 63° 26' 13" 


number — 0.009364 = log — 7.9714614 

R = 152 0 24' 37" = cot — 0.2818637 

x — 88° 58' 24" = cot * = + 8.2533261 





CHAPTER XXVIII. 


ADJUSTMENT OF PRIMARY TRIANGULATION. 

264. Method of Least Squares. —By the method of least 
squares is understood a process by which observations are 
adjusted and compared. When several precise measurements 
have been made of a given quantity, no matter how similar 
the conditions may appear, the results do not agree and it 
becomes necessary to adjust the various measures or observa¬ 
tions in order to get a mean or apparent agreement. The 
result is not necessarily the true value, but is used and 
accepted as such since it is a mean derived from the combina¬ 
tion and adjustment of all the measures taken which are most 
probably and apparently correct. 

Errors of observation are of two kinds, (1) systematic and 
(2) accidental; the former, resulting from unknown causes, 
affect all observations alike, while accidental errors are of a 
kind which produce discrepancies between observations: and 
it is this kind of errors alone, and not the systematic errors, 
which are considered in the so-called “theory of errors” and 
which it is the object of adjustment to minimize. The error 
of observation is truly the difference between the observed 
and true value, and may be plus or minus according as it 
exceeds or is less than the true value. The object of the 
theory of errors is to obtain from a number of discordant 
observations the best obtainable result. The fundamental 
principle of the method of least squares is the rule of Legendre, 
that, in observations of equal precision the most probable values 

602 


METHOD OF LEAS7’ SQUAEES. 603 

of observed quantities are those that render the sum of the 
squares of the residual errors a mininum. 

The probable value of an observed quantity is that which 
we are justified in considering as the more likely to be the true 
value than any other. As stated by Prof. Mansfield Merriman, 
the probability is expressed by an abstract fraction, which 
measures numerically the degree or likelihood in the happening 
or failing of an event; as confidence may range from improb¬ 
ability to certainty, so this measure may range from zero to one. 
If the figure 6, for example, occurs once on a die of six faces, 
the probability of its turning up when thrown is | ; likewise, if 
the same figure occurs on each face, the probability of its turning 
up when the die is thrown is f, or unity, which is certainty. 

When a number of unknown quantities are to be determined 
by means of equations involving unknown quantities, the 
quantities sought are said to be indirectly observed. It is 
necessary to have as many such indirectly observed equations 
as there are unknown quantities, and the discovery of these 
unknown quantities by solution of equations is the method of 
least squares. The differences between the several observed 
values and that which is taken as the true value of an obser¬ 
vation are called the residuals , and these are the apparent 
errors of observation. When observations are not made 
under the same conditions and the computer is aware of 
reasons which prevent them being equally good, a greater 
relative importance may be given to better observations by 
treating them as equivalent to more than one occurrence of 
the same value in a set of equal observations; in other words, 
they may be weighted. Weights (Art. 284) may therefore be 
regarded as numerical measures of the influence of the ob' 
servation upon the arithmetical mean. 

In observing a series of angles, the angles read at station 
Walton, between points n and 0 — a, 0 and p — b, p and q — c, v 
and q and r — d (Fig. 175, p. 613) are rendered functions of 
an adjustment equation. If combinations of these angles are 


604 adjustment of primary triangulation 

observed, as the angles between n and p — g, and between 
o and q — i, and between n and q = g -f- c, then the means of 
the various angles separately measured between n and o — a , 
and o and p = b, should be equal to the mean of the angle 
read between n and p — g. Similarly with the others, and by 
thus observing all of the separate and many of the combined 
angles it becomes possible to arrange a number of equations 
of condition , as they are called. In these, however, the mean 
observed angles never exactly sum up as they should theoreti¬ 
cally, and the differences are called the residuals. 

265. Rejection of Doubtful Observations.—When theod¬ 
olites or other angle-measuring instruments are used, there 
occur among a number of observations for the value of a par¬ 
ticular angle, one or more which differ greatly from the mean 
of all. It is not advisable to depend entirely on judgment as 
to which of these observations shall be retained and which 
rejected. The least objectionable criterion by which to judge 
as to the rejection of doubtful observations, and one based 
on mathematical principles, has been stated by Mr. T. A. 
Wright thus: 

Where an observation differs from the general run of the 
series by more than five times the probable error or three times 
the mean square error , attention should be called to it. 

An excellent fixed rule for the rejection of doubtful obser¬ 
vations is Peirce's Criterion , which is applied in the following 
manner: 

Let m — number of measures; 

n — number of doubtful observations to be rejected 
(to be found by trial); 

e = mean error of one observation in the set of m ; 
v y v', etc. = residuals of the observations or the difference of 

each value from the mean; and 
x = ratio of required limit of error for the rejection 
of n observations, to the mean error e\ so that 
xe is the limiting error. 


REJECTION OF DOUBTFUL OBSERVATIONS. 605 


The value of x 2 for n — 1, n = 2, etc., and for various 
values of m is found from Table XXXIV. All observations 
in which xe > v are to be rejected, stopping when xe for any 
value of n does not reject any observation for a value of n 
numerically one less. 


Example. 


No. 

Observed Angles. 

V 

*2 

1 

34 ° 

09' 

17".0 

o".o 

O 

2 

34 ° 

09' 

22".0 

5 "-o 

25.0 

3 

34 ° 

09' 

20'.0 

3 "-o 

9 0 

4 

34 ° 

09' 

16".5 

o "-5 

0.3 

5 

34 ° 

09' 

17".0 

o".o 

0.0 

6 

34 ° 

09' 

09". 5 

7"-5 

56.2 

Mean — 

34 ° 

09' 

17".0 

K© 

II 

90-5 


In the above column v contains the differences between 
consecutive observations, and the sum of the squares of the 
differences = 2v* = 90.5. This number divided by the 
number of observations less 1, or 




in — 1 


9 °- 5 

6 — 1 


= 18.1, 


gives a quotient which, multiplied by the number from the 
table of constants for six observations (Table XXXIV), gives 
2.592 X 18.1 = 46.9. Should any number in column v 1 
exceed this product, the observation from which it is found 
must be rejected. This rule requires the rejection of observed 
angle No. 6 and no other, and a new mean must now be 
found for the remaining angles, giving 34 0 09' 18A5. Peirce’s 
Criterion is also employed in determining the probable error 
and in rejecting doubtful observations in astronomic work. 




















6o6 ADJUSTMENT OF PRIMARY TRIANGULATION . 

Table XXXIV. 


PEIRCE’S CRITERION. 
Values of x 5 for 71 — i. 


m 

n 

i 

2 

3 

4 

O 

6 

4 

8 

9 

3 

i .480 


1 







4 

1.912 

[ >-163 








5 

2.278 

1 -439 








0 

2.592 

1.687 

M 

to 

0 

00 







7 

2.866 

1.910 

1.409 

* 045 






8 

3.109 

2.112 

1.589 

I .229 






9 

3-327 

2.295 

*■753 

i. 388 

1.09 r 





10 

2.526 

2.464 

1.904 

*• 53 * 

1.242 





11 

3-707 

2.621 

2.045 

1.662 

*•373 

1.122 




1*2 

3-875 

2.766 

2.176 

*•785 

1.492 

1.249 

1.018 



13 

4.029 

2 902 

2.299 

I .9OI 

1.604 

1.362 

*• *45 



14 

4-173 

3.030 

2.416 

2.029 

1.709 

1.465 

*•255 

*•053 


15 

4-309 

3 -> 5 1 

2.526 

2 . II I 

1.807 

1.561 

1-354 

1.163 


16 

4-436 

3.264 

2.63 j 

2.207 

1.898 

1.651 

*•445 

1.259 

1.080 

17 

4-555 

3 - 37 * 

2.729 

2.3OO 

1.985 

*•736 

*•529 

*-347 

1.176 

18 

4.668 

3-475 

2.824 

2.389 

2.069 

1.817 

1.609 

i. 428 

1.261 

19 

4.776 

3 - 57 i 

2.914 

2.474 

2.150 

1.895 

*•685 

*■ 5°4 

*• 34 * 

*20 

4.878 

3.664 

3-001 

2-556 

2.227 

1.970 

*•757 

*•576 

*■ 4*5 

21 

4-975 

3-755 

3.084 

2.634 

2.301 

2.041 

1.827 

1.644 

1.483 

22 

5.068 

3.840 

3.164 

2.709 

2.373 

2.109 

*•893 

I . 710 

*•549 

23 

5-157 

3-923 

3.240 

2.782 

2.442 

2.176 

*•957 

*-773 

1.612 

24 

5.242 

4.002 

3 - 3*5 

2.852 

2.509 

2.240 

2.019 

*•833 

1.671 

25 

5-324 

4.078 

3-387 

2.92O 

2.573 

2.302 

2.079 

1.892 

1.729 

26 

5-403 

4 - * 5 * 

3 - 45 6 

2.986 

2.636 

2.362 

2.137 

1.948 

1.784 

27 

5-479 

4.222 

3-523 

3-049 

2.697 

2.420 

2.194 

2,003 

1.838 

28 

5-552 

4.291 

3-588 

3.hi 

2.756 

2-477 

2.249 

2.056 

1.891 

29 

5.622 

4-358 

3-65* 

3 -* 7 * 

2.813 

2-532 

2.302 

2.108 

*• 94 * 

30 

5.690 

4.422 

3 - 7*2 

3.229 

2.869 

2.586 

2-354 

2.158 

1.990 

31 

5-756 

4.484 

3-772 

3-285 

2.923 

2.638 

2.404 

2.207 

2.038 

32 

5.820 

4-545 

3.829 

3-340 

2.976 

2.689 

2-454 

2.255 

2.085 

33 

5.882 

4.604 

3.884 

3-394 

3 -028 

2-738 

2.502 

2.302 

2.130 

34 

5-942 

4.661 

3-939 

3-446 

3.078 

2.787 

2-549 

2-347 

2175 

So 

6.001 

4 - 7*7 

3 - 99 2 

3-497 

3*27 

2.834 

2-594 

2.392 

2.218 

36 

6.058 

4 - 77 * 

4.044 

3-547 

3 - *74 

2.880 

2.639 

2.436 

2.261 

37 

6.113 

4-823 

4.095 

3-595 

3.221 

2.926 

2.683 

2.478 

2.302 

38 

6.167 

4.874 

4.144 

3-643 

3.267 

2.970 

2.726 

2.520 

2-343 

39 

6.2:9 

4-925 

4.192 

3.689 

3 - 3*2 

3013 

2.768 

2.561 

2383 

40 

6.270 

4-974 

4-239 

3-734 

3-356 

3-055 

2.809 

2.601 

2.422 

41 

6.320 

5.022 

4.285 

3-779 

3-398 

3-097 

2.849 

2.640 

2.460 

42 

6 • 369 

5-069 

4 - 33 * 

3.822 

3-440 

3.138 

2.888 

2.678 

2.497 

43 

6.416 

5-**4 

4-375 

3.865 

3.481 

3 - *78 

2.927 

2.716 

2-534 

44 

6.463 

5 - *59 

4.418 

3.906 

3.521 

3-217 

2.965 

2-753 

2-570 

45 

6.508 

5.202 

4.460 

3-947 

3-56i 

3-255 

3.002 

2-789 

2.606 

4(> 

6.552 

5-245 

4.501 

3-987 

3 600 

3-293 

3-°39 

2.825 

2.641 

47 

6.596 

5.287 

4-542 

4.026 

3-638 

3-330 

3-075 

2.860 

2.675 

48 

6.639 

5-328 

4.581 

4.065 

3-675 

3-366 

3.110 

2.894 

2.708 

49 

6.681 

5.368 

4.620 

4 -*03 

3 - 7*2 

3.40* 

3 - *45 

2.928 

2.741 

50 

6.720 

5 - 4 o 8 

4-657 

4.140 

3-748 

3-436 

3 - *79 

2.962 

2-774 

51 

6.761 

5-447 

4-695 

4.176 

3-784 

3-471 

3-213 

2-994 

2.806 

52 

6.800 

5.484 

4-732 

4-212 

3.819 

3-505 

3.246 

3.027 

2.838 

o3 

6.838 

5-522 

4.768 

4.247 

3-853 

3-538 

3-279 

3-059 

2.869 

54 

6.876 

5-559 

4.804 

4.282 

3.887 

3 - 57 * 

3-3** 

3-090 

2.899 

55 

6.913 

5-595 

4-839 

4.316 

3.920 

3.603 

3-342 

3 -* 2 I 

2.829 

56 

6.950 

5.630 

4-873 

4-349 

3-952 

3-635 

3-373 

3 -* 5 i 

2-959 

57 

6.986 

5.665 

4.907 

4.382 

3-984 

3.666 

3-404 

3.181 

2.988 

58 

7.021 

5-699 

4 - 94 1 

4 - 4*5 

4.016 

3-697 

3-434 

3.210 

3017 

59 

7.050 

5-733 

4-974 

4-447 

4.047 

3-728 

3-463 

3-239 

3.046 

60 

7.990 

5.766 

5.006 

4-478 

4.078 

3-758 

3-492 

3.268 

3-074 















































PROBABLE ERROR OF ARITHMETIC ME AH. 607 


266. Probable Error of Arithmetic Mean. —It is some¬ 
times desirable to determine the relative precision of different 
series of observations or their probable error. The probable 
error of the arithmetic mean of a number of measures is given 
by the formula 


R = 


0-6745 

Vm(tn — 1 ) 




in which R = probable error of arithmetic mean; 
0.6745 = a constant given by theory; 

2 = a symbol denoting “ the sum of.” 


The probable error of a single observation in the series is 
given by the formula 


r = 0.6745 





Example.—T he application of the foregoing formula is 
illustrated in the following tabular form : 


Between 

Observations. 


2 

3 

4 

5 

6 

7 

6 

9 

10 


Reduced Intervals. 


40° 35' 32".5 
31 .6 

26 .2 

25 .O 

25 .O 

27 -5 

28 .1 
18 .8 
20 .0 


Sum = 234". 7 
Mean = 26".077 


V 

v a 

+ 6".423 
+ 5 .523 
+ 0 .123 

- 1 .077 

- 1 .077 
+ 1 -423 

4 - 2 .023 

- 7 -277 

- 6 .077 

41".216 

30 .470 
.015 

I .166 

1 .166 

2 .Ol6 

4 .080 

52 .853 

36 .966 

= 169' .948 

-2V 2 169".948 „ 

= Q = 2 l ".243 

m — 1 8 

„ 0.6745 _ 

R — — - V 2 v *= ± i".04i 

V ? n(tn — 1 ) 























608 ADJUSTMENT OF PRIMARY TRIANGULATION . 


The factors 
low. 


0-6745 

V m(m — 1 ) 


and 


0-6745 

Vm — 1 


are tabulated be- 


Table XXXV. 

FACTORS FOR COMPUTING PROBABLE ERROR BY BESSEL’S 

FORMULAS. 


m 

Single 

Observations. 

Set of 

Observations. 

0.6745 

0.6745 

- 1 

^ m(ni — 1) 

2 

.6749 

.4769 

3 

.4769 

•2754 

4 

.3894 

.1947 

5 

•3372 

. 1508 

6 

.3016 

.1231 

7 

•2754 

. IO41 

8 

•2549 

.0901 

9 

.2385 

•0795 

10 

. 2248 

.0711 

11 

.2133 

.0643 

12 

• 2034 

.05S7 

13 

.1947 

.0540 

M 

. 1871 

.0500 

15 

. 1803 

.0465 

16 

.1742 

•0435 

*7 

.1686 

.0409 

18 

• 1636 

.0386 

19 

.1590 

.0365 

20 

•1547 

.0346 


267. Reduction to Center. —The first operation in the 
computation of a system of triangulation is that of reducing 
to the center of the station such observations as were taken 
with the instrument not centered over it. The mode of mak¬ 
ing such reduction is best illustrated by the following example 
taken from the triangulation of the U. S. Geological Survey 
in Kansas, in which the position of the instrument on the 
station Walton was eccentric to the station. 
























REDUCTION TO CENTER. 



In Fig. 174 let 

P = place of instrument; 

C = center of station; 

O — angle at P between two ob¬ 
jects, A and B ; 

y = angle at P between C and 
the left-hand object, B; 

r = distance CP; 

C — unknown and required angle 
at C; 

D — distance AC; 

G — distance BC ; and 

A — angle at A between/* and C. 



Fig. 174.— Reduction to Center. 


hence 


Then, from the relation between the parts of the triangle, 

G : r : ; sin y : sin B; . . . . (64) 


. r sin y 

sin B = —^—. 


• (65) 


As the angles at A and B are very small, they may be re¬ 
garded as equal to A sin 1" and B sin 1" ; hence 




B = (in seconds) 


r sin y 
6 sin 1"’ 


. . . ( 66 ) 


and 


C=0 + r - Z {0 % y) -P^ n . . . (67) 

1 ) sin 1 G sin 1 7 


In the use of this formula proper attention should be paid 
to the signs of sin (0 -J- y) and sin y ; for the first term will 
be positive when {0 + y) is less than 180° (the reverse with 
sin y ); D being the distance of the r^///-hand object, the 
graduation of the instrument running from left to right. 

r being relatively very small, the lengths of D and G are 
approximately computed with the angle 0 . 

The folloiving quantities must be known in addition to the 






6 IO ADJUSTMENT OF PRIMARY TRIANGULA TION. 

measured angles in order to find the correction for reducing 
to center: 

1. The angle measured at the instrument, P, between the 
center of the signal or station, C , and the first observed station 
to the left of it, B. 

2. The distance from the center of the instrument to the 
center of the station = r. 

3. The approximate distances, D, G, etc., from the sta¬ 
tion occupied to the stations observed. The latter may be 
computed from the uncorrected angles. 

The practical mode of determining the correction to each 
angle read at the instrument on Walton to a corresponding 
angle at the center of the signal or station (Fig. 175) is illus- 
strated as follows: 

Example.—Reduction to center of station at Walton 

(See explanation : Appendix No. 9, page 167, U. S. Coast and Geodetic 

Survey report for 1882.) 

Distance, inst. to center = o'.48 log = 9 6812; 

log feet to meters = 0.5160; 

Distance, inst. to center log meters = 9.1652 = log r . 


Direction. 

x to n 
7 °. 

x to 0 
73 °- 

x to p 
105°. 

x to q 
185°. 

x to r 
273 °- 

X to s 
306°. 

I, 

log sin angle. 

9.0859 

9.9806 

9.9849 

8.9403 

9.9994 

9.9080 

a.c. log, distance. 

5-9321 

5 - 9 T 82 

6.4228 

6.2434 

6.OO79 

6.2514 

log r . 

9.1652 

9 -I 652 

9 -I 652 

9 -I 652 

9.1652 

9-1652 

a.c. log sin 1 ". 

5 . 3 M 4 

5-3144 

5 - 3 M 4 

5-3144 

5-3144 

5.3144 


9.4976 

O.3784 

O.8873 

9.6633 

O.4869 

0.6390 

Correction to direction.. 

o"3i 

2".39 

7".71 

o ".46 

3".06 

4 "-36 

L- 








// // // 


Correction to angle a = 

n 

to 0—0.31+2.39 = 

+ 2.08 

Check 2' 

'.08 

8" 

• 17 

b = 

0 

to /-2.39+7.71 = 

+ 5-32 

5 

•32 

2 

.60 

S = 

n 

to/-0.31+7.71 = 

+7.40 

4 

.67 

1 

.30 

c = 

P 

to ^ — 7.71—0.46 = 

— 8.17 





d - 

I 

to r+0.46 — 3.06 = 

— 2.60 

12 " 

.07 = 

12" 

.07 

e = 

r 

to J+3.06 — 4.36 = 

— i- 3 ° 


.1 



h - 

I 

to s -}-o. 46— 4* 36 = 

- 3-90 





/ = 

s 

to w-t--4.36-fo.31 = 

d-4.67 


































ST A TION AD JUS TMEN T. 


6 l I 

The corrections 2'.o8, -f- 5 -3 2 > etc., found in the last 
column above, are those which are applied to the observed 
angles (Example, Art. 268) to reduce them to center of sta¬ 
tion. 

268. Station Adjustment. —Doubtful observations hav¬ 
ing been eliminated and the observed angles having been 
reduced to center of station, the next step is the station 
adjustment. The sum of all the angles closing on the 
horizon and observed at the center of any stations should 
equal 360°, and the sum of any two angles, as a and b (Fig. 
175), should equal their combined observed angle ^ In fact, 
it will be found that this is not the case owing to errors of 
observation due to various causes (Chap. XXVI). The object 
of the station adjustment is to so distribute these errors among 
the angles a, b, and g as to give the most probable values 
which will satisfy these conditions. The following example 
is taken from the same station Walton, as is the example of 
reduction to center (Art. 267). 


Example. 



Obs 

Angles. 

Station 

Adjust¬ 

ment. 

Reduc¬ 
tion to 
Center. 

Angles Locally 
Adjusted and 
Reduced to 
Center. 


O 

/ 

// 


// 

// 

O 

/ 

// 

a Dunkard—Peabody. 

65 

45 

28.37 

4 - 

•51 

—j— 2.08 

65 

45 

30.96 

b Peabody—Newt. 

3 1 

47 

58.50 

4 - 

• 52 

+5-32 

3 1 

48 

04-34 


07 

77 

26.87 




07 

77 

7«c. 70 

g Dunkard—Newt (meas.). 

97 

33 

28.39 

— 

• 49 

+7-40 

97 

33 

35-30 


- i- 5 2 






OO .00 








d Township cor.—Royer. 

87 

44 

57.41 

_ 

•56 

— 2.60 

87 

44 

54-25 

e Royer—Bennett. 

34 

OO 

03.35 


• 5 6 

—1.30 

34 

OO 

01.49 


I 21 

44 

60.76 




121 

44 

55.74 

h Township cor.—Bennett. 

121 

44 

59-05 

+ 

•59 

— 3 90 

121 

44 

55-74 


4 - 1.7* 




OO. OO 

J Bennett—Dunkard. 

6l 

09 

26.17 

+ 

.02 

4- 4.67 

6l 

09 

30.86 

g Dunkard—Newt. 

97 

33 

28.39 

— 

• 49 

4 - 7-40 

97 

33 

35-30 

t Newt — Township cor. 

79 

32 

06.25 

+ 

.02 

- 8.17 

79 

31 

58.10 

h Tp. corner. — Bennett. 

121 

44 

59-05 

4 

•59 

- 3 - 9 o 

121 

44 

55-74 


359 

qn 

eq. 86 




360 

00 

00.00 



— 0.14 




00.00 





















































61 2 AD JUS TMEN T OF PRIM A R Y TRIA NG ULA I'ION. 

269. Routine of Station Adjustment. —In the solution of 
a station adjustment a certain fixed routine is followed which 
furnishes the simplest arrangement for determining unknown 
corrections to the angles read around the horizon. The 
various operations performed in the course of this solution are 
elaborated in the following articles; some are identical with 
those performed in the solution of a figure adjustment (Art. 
273), to which latter operation reference is made in the proper 
places. The routine consists of the following: 

1. The determination of the differences between separate 
observed angles and their combined observed angle as between 
a -f- b and g , as shown in the second column of the preceding 
example. These furnish the equations of condition. (Art. 
270.) 

2. The formation of a table of correlates from the equa¬ 
tions of condition. (Art. 271.) 

3. The transfer of the table of correlates into normal 
equations. (Art. 272.) 

4. The solution of the normal equations for the determi¬ 
nation of the unknown quantities. (Arts. 272 and 281.) 

5. The substitution of the corrections found back into 
the table of correlates. (Art. 272.) 

6. The placing of the corrections to the angles found by 
the last operation in the proper place. (Example, Art. 268, 
column three, also Art. 272.) 

7. The addition or subtraction of the corrections to 
or from the observed angles. (Example, Art. 268, column 
three.) 

'8. The addition or subtraction of the correction resulting 
from reduction to center (Art. 267) to or from the corrected 
observed angles. (Example, Art. 268, column four.) 

270. Equations of Condition. —In the column of “Ob¬ 
served Angles” (Example, Art. 268) occur the following three 
equations of condition : 


EQUATIONS OF CONDITION. 613 

(A) a + b — g — 1" . 52 =0; 

(B) d + e - Ji+ 1". 71 =0; 

(C) f + g + c + h — o" . 14 = 0; 

in which the letters represent not the angles, as in the dia¬ 
gram, but unknown corrections to the angles. The method 
of solving these equations is briefly described in the next 
Article. The description is elaborated in the example of a 
figure adjustment (Arts. 273 to 284), which may be consulted 
in this connection. 

The number of equations of condition which may be 
arranged is limited only by the number of single and com¬ 
bined angles observed (Fig. 175). In fact, however, provid- 


71 



Fig. 175.— Station Adjustment. 

ing all of the possible combinations have been observed, only 
a portion of them are employed in the adjustment and used 
to make additional conditions. This number is limited by 
the angles used which enter into the figure adjustment (Arts. 
276 and 278). Thus it is unnecessary to introduce conditions 
by the adjustment of angles which will not form a part of 
some one of the figures which is to be adjusted later. In the 
case considered the figures resting on the station Walton are 




614 adjustment of primary triangulation . 

so disposed about it that it is necessary to solve but three 
equations of condition. 

The above equations result from the fact that the sum of 
the angles a and b , as separately observed, fails to equal their 
corresponding sum angle g, as it was measured. The differ¬ 
ence 1A50 is the amount which is to be divided among the 
three observed angles as a correction to each. The object of 
the least-square station adjustment is to so apply these 
corrections that the resulting angles will be such that 
a + b — g — o; and so for the others. 

271. Formation of Table of Correlates. —By this mode 
of solution corrections are found which fulfill the conditions 
expressed in the equations of condition (Art. 270). The 
method of least squares is described in Article 264, and no at¬ 
tempt will be made to explain it theoretically here. The lists 
of works of reference (p. 809) show where the theory may be 
studied, the more important books on the subject being those 
of Chauvenet, Wright, and Merriman. Its application in the 
simpler geodetic operations is best explained by examples, of 
which this is typical. 

In the solution of the above equations by the method of 
least squares, they are first written in the form of a table of 
correlates, the letters at the top designating the equations, as 
follows: 



A 

B 

C 

a 

1 



b 

1 



c 



1 

d 


1 


e 


1 


f 



1 

g 

— 1 


1 

h 

. _ 


— 1 

1 


Thus a occurred in equation (A), now called column A, 
+ I time, and the figure 1 is written opposite a in column A. 
So ^occurs — 1 time in equation (A), and is written in column 










FORMATION OF NORMAL EQUATION. 61 5 

A; also -(- 1 time in equation (C), and is written so in column 
C opposite g, etc. 

The above table of correlates is now solved according to 
the algebraic formula: 

a 1 + ab -f- ac -f- ad, etc. ; >. 

ba —|— U 1 -|— be —j- bd, etc. ; v . . . . (68) 

ca -f- cb -f- c -f- cd, etc.; ' 

272. Formation of Normal Equations and Substitution 
in Table of Correlates. —-This is accomplished by multiply¬ 
ing each coefficient in the above table by itself and by every 
other in the same horizontal line and summing them. Then, 
by substituting the result back in the equations of conditio^, 
there are formed the following three normal equations. Thus 


A. + 3.00A.— 1.00C — i".52 = o 

B .-f- 3.00B — 1.00C -j- i".7i = o 

C. — 1.00A — 1.00B -f- 4.00C — o". 14 = o 


column A is multiplied vertically into itself 3 times; into B, 
horizontally, no time, as neither column has coefficients on 
the same horizontal line; and into C, horizontally, once. 

These three equations, involving three unknown quantities, 
are then solved by elimination (Art. 281) with results as fol¬ 
lows : 

n 

A = + -5 l S 5 

B = — .562 ; 

C = + .023. 

These values of A, B, and C can now be substituted in 
the table of correlates (p. 164), columns A, B, and C; the 
algebraic sum of lines a , b, c , d, etc., giving corrections to the 
angles a, b, c, d, etc. 





6 l6 ADJUSTMENT OF PRIMARY TRIANGULATION. 



A 

B 

C 

Corrections 
to Angles. 

a 

+ -515 



// 

T • 5 I 5 

b 

T • 5 I 5 



-+- -515 

c 



-f .023 

+ .023 

d 


— .562 


— .562 

e 


- .562 


— .562 

f 



+ .023 

+ .023 

or 

£> 

- - 5 i 5 


H- -023 

- -492 

h 


+ -562 

+ .023 

+ .585 


The above are the corrections which have been entered in 
the example (p. 61 i) under the column heading “Station 
Adjustment.” The algebraic summation of the corrections in 

i 

that column and those in the column headed “ Reducction to 
Center” give the column of final locally adjusted angles. In 
these it will be noted that the sums of the various observed 
angles exactly equal their corresponding combined observed 
angle. 

273. Figure Adjustment. —In primary triangulation com¬ 
putation for figure adjustment means the fulfilling of the con¬ 
ditions imposed by the various triangles which form geometric 
figures. (Art. 238.) The length of any side in any triangle 
in a triangulation net being known and all the angles measured, 
the length of any other side may be computed by following at 
least two independent routes through the intervening triangles. 

The object of a figure adjustment is to find from a given 
set of measured angles the values which will remove the 
contradictions among them and will satisfy the following two 
classes of conditions: 

1. The local conditions , or those arising at each station 
from the relations of the angles to one another at that station. 
These are satisfied by the station adjustment. (Art. 268.) 

2. The general conditions, or those arising from the geo¬ 
metrical relations necessary to form a closed figure. These 
are satisfied by the figure adjustment, and are of the following 
three kinds: 













FIG U RE AD JUS TMEN T. 617 

(a) The sum of the angles of each triangle must be equal 
to 180° plus spherical excess. 

(b) The length of a side must be the same by whatever 
route it is computed from the given base. 

(c) The adjusted values of the angles must be the most 
probable that can be found from the observations. 

Ordinarily the angles have been measured by instruments 
and methods better than the requirements of mapping, and 
in such cases it is not necessary to make a-figure adjustment 
other than an arbitrarily equal or perhaps weighted distri¬ 
bution of the error of each triangle among the three triangles 
which compose it. The necessity of a more elaborate adjust¬ 
ment may arise where the computations are to be carried 
through a long scheme of triangulation connecting distant 
points, when it becomes desirable to make so rigid an adjust¬ 
ment that any connection with this scheme of triangulation 
from any direction will not alter the computed quantities. 

Rigid figure adjustment is made by the method of least 
squares, and the simplest mode of explaining such adjustment 
is by an example taken from actual practice rather than by 
algebraic formulas. The latter may be found fully elaborated 
by Chauvenet, Merriman, etc. Such an example is the 
following, taken from a scheme of triangulation executed 
with vernier theodolite for the U. S. Geological Survey and 
computed and in part elaborated by Mr. E. M. Douglas: 

274. Routine of Figure Adjustment. —In making a 
figure adjustment a certain tabular routine is followed, be¬ 
cause it furnishes the simplest arrangement of solving a 
complicated series of algebraic problems and in the most 
mechanical manner. This consists practically of seven sepa¬ 
rate operations, which are elaborated in the following articles. 
The order presented is, after a description of the notation, as 
follows: 

1. The formation of the angle equations. (Art. 276.) 

2. The determination and application of the spherical 
excess to each triangle. (Art. 277.) 


6 l8 ADJUSTMENT OF PRIMARY TRIANG ULATION. 


3. The formation of side equations. (Art. 278.) 

4. The solution of the angle and side equations, which is 
performed as one operation. (Art. 279.) 

5. The summation of angle and side equations (Art. 279), 
which consists of the following separate operations: 

(a) The formation of a table of correlatives. (Art. 280.) 

(< b ) The formation of normal equations from the table of 
correlatives. (Art. 280.) 

(c) The algebraic solution of the normal equation by the 
least-square method. (Art. 281.) 

(d) The substitution back into the normal equation of 
the values found by elimination. (Art. 282.) 

(e ) Solution of the table of correlatives whereby the 
numerical values of the corrections to the angle and to the 
sides are obtained. (Art. 283.) 

6. The substitution of the corrections to the angles and 
sides back into the angle and side equations (third and fourth 
columns, example, Art. 276, and fourth and fifth columns, 
example, Art. 279). 

7. The determination of the final corrected spherical 
angles and sides as a result of the application of the side and 
angle corrections to the observed angles (last column of 
examples, Arts. 276 and 279). 

275. Notation Used in Figure Adjustment. —An angle 
may be considered to be the difference in azimuth or direc¬ 
tion of two lines bounding it. Azimuths 
are counted from the south through the. 
west, north, and east. Therefore the 
angle 4.1.2 is equal to the azimuth of 
the line 2.1 minus that of 4.1, and 
may be written 

4.1.2 or — f + f. 

In the second form the number at the 
vertex of the angle is always written 
underneath the other numbers. If written in the first man- 



Side Notation. 


ANGLE EQUATIONS. 


619 


ner, 4.1.2, the numbers should be in such order that when 
the vertex of the angle is toward the observer the left-hand 
station number is written first. The angle 4.1.2, if written 
without signs, would then be read minus the side (or direc¬ 
tion) 4.1 plus the side 2.1; that is, the angle would be 
— 4.1 + 2.1. 

276. Angle Equations. —The sums of the three measured 
angles of any triangle should = 180 0 -f- its spherical excess. 
Each fails to do this, however, by a small amount, which is 
distributed as a correction to each angle. As a result each 
triangle furnishes an equation of condition , which is called 
the angle equation. The number of angle equations in any 
figure is equal to the number of closed sides in the figure -f- I 
and — the number of stations. Thus in a closed quadrilateral 
(Fig. 177) the number of angle equations is 6 + 1 — 4 == 3. 
The corrections to the angles as found by solution are in¬ 
serted in the fourth column of the following example of a 
figure adjustment. In this example the various angles are 
designated in the first column, in accordance with the nota¬ 
tion just given; in the second column are written the plane 
angles resulting from the station adjustment (Art. 268) and 
the accompanying correction for reduction to center; in the 
last column are given the corrected spherical angles, the sum 
of which must equal i8o°+ spherical excess. 

277. Spherical Excess. —The angles observed in the field 
are measured on a spherical surface, and the sum of the three 
measured angles of each triangle should, if exactly measured, 
equal 180° plus spherical excess. This quantity must be 
computed and subtracted from the sum of the angles only for 
the purpose of testing the accuracy of closure of the triangle, 
since in the final computation the angles are treated as plane 
angles. 

Since the spherical excess amounts, between latitudes 25 0 
and 45 0 , to about 1" for an approximate area of 75.5 square 
miles, an empirical formula for approximately determining 


620 ADJUSTMENT OF PRIMARY TRIANG ULA TION. 


Example.— ANGLE EQUATIONS. 


Triangle 

Sides. 

Observed Angles. 

Corrections to Sides. 

Corrections 
for each 
Angle. 

Corrected 
Spherical Angles. 

1. 2. 3. 

2. 3. I. 

3. I. 2. 

O / // 

123 26 46.67 

34 09 II.89 

22 23 54.34 

II II 

+ I.447 + 0.962 
+ 0.960 + 1.534 
+ 1-538 + 1.448 

// 

+ 2.41 
+ 2-49 
+ 2.99 

O III 

123 26 49.08 

34 09 14.38 

22 23 57.33 

Sum = 

a . 

179 59 52.90 
s. e. 0.79 

Error, — 7.89 

+ 7*889 

-j- 7.89 

/ 

180 00 00.79 

2. 3- 4- 

3. 4. 2. 

4. 2. 3. 

76 57 41.76 

45 15 50.41 

57 46 28.77 

4- 0.960 — 0.574 

- 0.575 - 0.487 

— 0.485 + 0.962 

-f O.38 
— I.06 

+ O 48 

76 57 42.14 

46 15 49.35 

57 46 29.25 

Sum = 

• • • • • • 

180 00 00.94 
s. e. 0.74 

Error, + 0.20 

— 0.199 

— 0.20 

180 00 00.74 

1. 3. 4. 

3. 4. 1. 

4 . 1. 3. 

42 48 29.87 
105 36 13.12 

31 35 22.54 

— 1.534 - 0.574 

— 0.575 — 0.088 

— 0.090 — 1.538 

— 2 .II 

— 0.66 

— 1*63 

42 48 27.76 

105 36 12.46 

31 35 20.91 

Sum = 

c . 

180 00 05.53 
s. e. 1.13 

Error, -f- 4.40 

- 4-399 

- 4.40 

180 00 01.13 

1. 2. 4. 

2. 4. 1. 

4. 1. 2. 

65 40 17.90 

60 20 22.71 

53 59 16.88 

+ 1.447 -f 0.485 
-j- 0.487 — 0.088 
— 0.090 -f- 1.448 

+ 1-93 
-J- 0.40 
+ 1.36 

65 40 19.83 

60 20 23.11 

53 59 18 24 

Sum = 

179 59 57-49 
s. e. 1.18 

Error, — 3.69 

+ 3-689 

+ 3-69 

1 0 00 01.18 













































































SPHERICAL EXCESS. 


621 


spherical excess in triangles of less area than 500 square 
miles is 


E (in seconds) = 


area sq. mi. 

75.5 


at lat. 40°. 



For latitude 20° a constant divisor is 74.76, and for lati¬ 
tude 6o° it is 76.42. The area of the triangle may be com¬ 
puted with sufficient accuracy by considering the angles as 
correct, and subtracting one-third of the excess of the angles 
above 180 0 from each angle. 

When the area of a triangle is larger than 100 square 
miles the spherical excess in seconds should be determined 
by the equation 

A ab sin c 

JE* — 2 • / / ' *1 • / / y 

r sin 1 2 r sin 1 



in which A — area of triangle in square miles, and 

r — radius of curvature of the earth in miles, and 
is a constant for a given latitude, or may be assumed as a 
constant in the latitudes included within the area of the 
United States. 

, r „ ab sin c , , , , 

The value of A — --- may be determined by the 

empirical formula (69). The log mean radius of earth in 
miles, r — 3.5972790. 

As the value of the divisor in formulas (70) is a constant 
for different latitudes, it may be expressed thus: 


m — 


2 r* sin 1" ’ 


and we have 


E — ab sin C X w. • 


• • (71) 







622 ADJUSTMENT OF PRIMARY TRIANGULA IT ON. 

Table XXXVI. 

LOG m FOR DETERMINING SPHERICAL EXCESS. 

FOR DISTANCES IN METERS. 

(Computed for Clarke’s Spheroid of 1866 from Appendix 7, U. S. Coast 
and Geodetic Survey Report for 1884.) 


f 

Latitude. 

Log m . 

Latitude. 

Log m . 

Latitude. 

Log m . 

Latitude. 

Log tn. 

O. / 

20 OO 

I.40625 

O / 

32 OO 

I.40528 

O / 

44 00 

I.404IO 

O / 

56 OO 

I.40290 

20 30 

622 

32 30 

524 

44 30 

405 

56 30 

285 

21 00 

619 

33 00 

519 

45 00 

400 

57 00 

280 

21 30 

615 

33 30 

514 

45 30 

395 

57 30 

276 

22 00 

612 

34 00 

509 

46 00 

390 

58 00 

271 

22 30 

608 

34 30 

505 

46 30 

385 

58 30 

266 

23 00 

604 

35 00 

500 

47 00 

380 

59 00 

262 

23 30 

601 

35 30 

495 

47 30 

375 

59 30 

257 

24 00 

597 

36 00 

491 

48 00 

369 

60 00 

253 

24 30 

592 

36 30 

486 

48 30 

364 

60 30 

249 

25 00 

588 

37 00 

481 

49 00 

359 I 

61 00 

• 244 

25 30 

584 

37 30 

476 

49 30 

354 

61 30 

240 

26 00 

580 

38 00 

47 i 

50 00 

349 

62 00 

236 

26 30 

576 

38 30 

466 

50 30 

344 

62 30 

231 

27 00 

572 

39 °o 

461 

51 00 

339 

63 00 

227 

27 30 

568 

39 30 

456 

5130 

334 

63 30 

223 

28 00 

564 

40 00 

45 i 

52 00 

329 

64 00 

219 

28 30 

559 

40 30 

446 

52 30 

324 

64 30 

215 

29 00 

555 

41 00 

441 

53 00 

319 

65 00 

211 

29 30 

55 i 

41 30 

436 

53 30 

314 

65 30 

207 

30 00 

547 

42 00 

43 i 

5 4 00 

309 

66 00 

203 

30 30 

542 

42 30 

426 

54 30 

304 

66 30 

200 

31 00 

537 

43 00 

420 

55 00 

299 

67 00 

196 

31 30 

_ 

1 -40533 

43 30 

1.40415 

55 30 

1.40295 

67 30 

I.4OI92 








































SIDE EQUATIONS. 


623 


Example.— Let a and b be the lengths of the two sides, 
and L the included angle; in is a constant to be derived 
from Table XXXVI for distances in meters. For the mean 
latitude 30° 40' of the example chosen, m — 1.40540. Then 
we have, solving by formula (71), 


Triangles 

1-2.3 

2 3-4 

1 - 3-4 

1.2.4 

Angles C — 

123 0 26' 50" 

76° 57' 40" 

42 0 48' 30" 

65° 40' 20" 

Log (side) a 
“ “ b 

“ sine C 

“ const, m 

4.36885 

4-20055 

9.92137 

1.40540 

4-20055 

4.27642 

9.98866 

1.40540 

4-54093 

4.27642 

9.83222 

1.40540 

4.36885 

4-33733 

9.95962 

1-4054° 

Log s.e. = 
s.e. = 

9.89617 

0" - 79 

9 - 87 i0 3 

o"-74 

0.05497 

1". 13 

0.07160 

T'.i8 


Then one-third o".yg is to be subtracted from each of the 
three angles of the triangle 1.2.3, etc. 

278. Side Equations. —It is evident that the distribution 
of corrections to the three angles of a triangle to make their 
sum 180 0 , affects not only the angles, but as a consequence 
the sides, diminishing the lengths of the latter as the angles 
are diminished or increasing the lengths as the opposite 
armies are increased. Therefore the adjustments to the sides 

o 

of the triangles must be made with the adjustments to the 
angles in order that the triangle shall not be distorted. The 
determination of the corrections to be applied to the sides is 
performed through the formation of side equations, which are 
best explained by reference to Fig. 177. The solution of 
these is performed at the same time with that of the angle 

equations in Articles 279 to 283. 

Suppose 4.1.2.3 to represent the projection of a pyramid 
of which 1.2.3, the shaded side, is the base and 4 the apex. 























t> 24 ADJUSTMENT OF PRIMARY TRIA NG U LA T 10 N. 

A geometric condition of such a figure is that the sums of the 
Logarithmic sines of angles about the base taken in one direction 


4 



Fig. 177.—Angle and Side Equations. 


must equal similar sums taken in the other direction ; that is ; 
the product of the sines must be equal. In this case, therefore, 

log sine 4.1.2 + log sine 4.2.3 + log sine 1.3 4 
should equal 

log sine 1.2.4+ log sine 2.3.4 + log sine 4.1.3. 

The number of side equations which can be formed in any 
figure is equal to the number of lines in the figure plus three, 
minus twice the number of stations, or 1 + 3 — 2 n. In a 
quadrilateral, therefore, 6 + 3 — 8 = 1 ; hence such a figure 
contains one side equation, or equation of condition. The 
numerical term in each side equation is the difference between 
the sums of the logarithmic sines taken in each direction. 
The coefficients for the unknown corrections are the differences 
for one second in the logarithmic sines of the angles. 

Further examples of the method of arranging equations of 
condition and applying corrections to the angles may be best 
shown by a continuation of the example selected (Art. 276), 
which is the simplest, being that of a quadrilateral (Fig. 177^ 





SOLUTION OF ANGLE AND SIDE EQUATIONS. 625 

The method of forming correlative and normal equations 
and their solution is similar to that for station adjustment 
(Arts. 271, 272). \x\ \\\z equations of conditions and correla¬ 

tives the angles are designated by directions to which the 
corrections are finally applied. 

279. Solution of Angle and Side Equations. —The cor¬ 
rections to the sides bounding an angle are empirically denoted 
by enclosing them in brackets and prefixing the proper signs. 
Thus the corrections to be found for the angle 3.4.1 may be 
denoted by its side corrections, — (f) -j- (J). 

In the triangle 1.2.3 write each observed angle and indi¬ 
cate a correction, to be found, as above: 

1.2.3 - [ 1 - 2 ] + [ 3 - 2 ] + 2.3.I - [2.3] + [1.3] + 3.I.2 

— [3- 1 ] + [2.1] =180 + s.e.; 

or, instead of the angle numbers, 1.2.3, etc.,write their com¬ 
bined value: 

179 ° 59 ’ 5 2 "-9 - [1.2] + [ 3 - 2 ] - [2.3] + [1.3] - [ 3 -i] 

-f- [2.1] == 180 0 + s. e. = 180 0 oo' 00".79. 

This equation reduced gives 

— [ 1 *2] { [3 • 2] [2.3] 1 [ i* 3 ] [ 3 * 1 ] I [2- 1] 7 -89 = o. (a) 

In this manner equations are formed for the other triangles, 

1.2.4 being assumed to be the dependent triangle, and we 
have: 

-[2-3]+[4-3]-[3-4]+[2.4]-[4-2]+[3-2]+o.20 = o; . (6) 

-[i-3]+[4-3]-[3-4]+[i-4]-[4-i]+[3-0+4-40=0. . (c) 

Arrange the logarithmic sines as shown in the following 
tabular example by writing opposite each sine its logarithmic 
difference for one second as given in a table of seven-place 
logarithms. The logarithmic differences for angles greater 


626 ADJUSTMENT OF PRIMARY TRIANGULATION. 


than 90° will have a minus sign, for those less the sign will 
be plus. For a small correction to any of these angles the 
change in the logarithmic sine will be equal to the correction 
in seconds multiplied by the tabular difference for 1 ". As 
the correction for the angle 3.4.1 is denoted by — (f) -f- (J), 
applying the tabular difference for 1", we have the change in 
the log. sine — 5.9 (— (f) + (J)). 


Example.—Summation of Side and Angle Equations. 


■ 

Angles. 

Log. Sines. 

Difference, 

1". 

Correc. to 
Angles. 

Correc. to 
Sines. 

Corrected 

Sines. 

3 . 4 , I 

3 . 1. 2 

4, 2, 3 

9.9836918.6 

9.5809762.8 

9.9273484.8 

- 5-9 
4-51.1 

+ 13-2 

// 

— 0.66 
+ 2.99 
+ °-48 

+ 3-9 
+ 152.8 
+ 6.3 

9.9836922.5 

9.5809915.6 
9.9273491.I 

Sum... 

+ 

9.4920166.2 



+ 163.0 

O . ACi 9 C \' X 9 >Ci 9 




4 . 1, 3 
1, 2, 3 

3 . 4 . 2 

9 . 7 I 9 I 9 I 3-9 

9.9213757.3 

9.8514769.5 

+ 34-2 
— 13-9 
— j- 20 . 8 

— 1.63 
+ 2.41 

— 1.06 

-- 55-7 

- 33-5 

— 22.0 

9.7191858.2 

9.9213723.8 

9.8514747-5 

Sum... 

9.4920440.7 
9.4920166.2 

1 


— in. 2 

9.4920329.5 




Error = - 274.5 



274.2 







■ ■ ■ ■■ -J 


As shown in the beginning of this Article, the equation 
which must be satisfied in an adjusted quadrilateral to fix the 
relative length of the sides is 

log sin 3.4.1 -|~ log sin 3.1.2 log sin 4.2.3 

— log sin 4.1.3 — log sin 1.2.3 — log sin 3.4.2 = o. 

Substituting in this, with changed signs, corrections 
found as above, we have, after reducing, the following side 
equation : 





















































CORRELATES AND NORMAL EQUATIONS. 627 

+ 5-9C3-4] - 5-9L 1 - 4 ] - 5i-*[3-i] + 5 1• i[ 2 .i] - 13 . 2 I+ 2 ] 
+ i3-2[3.2] + 34-2[4-i] - 34-2[3-i] - i3-9[ JC - 2 3+ 1 3-9C3-2] 
+ 20.8[3.4] — 20.8[2.4] — 274.5 = o. 

This being a true algebraic equation, it may be divided 
by any number without altering its value. Dividing it through 
by some convenient multiple of 10, as 80, to give smaller 
coefficients, and combining algebraically the coefficients of 
[3.4], [3.2], [3.1], each of which appears twice, gives 

+ - 334 [ 3 - 4 ] - .Q 74 [i. 4 ] - i.o66[3.i] +.639(2.1]-i.6s[4.2] 
+- 339 [ 3 - 2 ]+. 427 [ 4 *i]--i 74 [i. 2 ]-. 26 o[ 2 . 4 ]- 3.427 = o. (d) 
Thus for example, for the coefficients of [3.4], we had 
+ 5-9 [34] + 20.8 [3.4] = +26.7 [3.4] 80 = .334 [3.4], 

and so for the others. 

280. Correlates and Normal Equations. —We now have 
the equations necessary for the complete adjustment of the 
quadrilateral, and from them values must be found by means 
of correlates, each equation of condition having a correlate, 
and each correction coefficient giving a correlate coefficient. 
The algebraic sum of the coefficients of each correlate will be 
zero. 

The following tables of correlates can now be formed from 
the above, as was done in the station adjustment (Art. 271). 
The solution of this can only be made after the normal equa¬ 
tion has been formed and solved by elimination (Art. 281). 

The first of the tables is formed from the adopted equa¬ 
tions of condition (a), (&), (c), and (d) (pages 625 and 627) by 
arranging corresponding columns a, b, etc. In these and on 
line with the various side numbers 2.1, 3.1, etc., are placed 
the coefficients of the latter with their signs from the equa¬ 
tions (a), (b), etc. Thus on line 1 the side 2.1 occurs + 1 
time in equation (a) and again + .639 times in equation (d), 
and so for the other sides. 


628 ADJUSTMENT OF PRIMARY TRIANGULATION. 


EXAMPLE. 

TABLE OF CORRELATES TABLE OF CORRELATES 

FORMED. SOLVED. 


Line. 

Sides. 

a . 

b. 

c . 

d . 

I 

2.1 

+ 1 



4 - -639 

2 

3-t 

— 1 


" 4 ” i 

— 1.066 

3 

4.1 



— I 

4 - .427 

4 

1.2 

- 1 



- -174 

5 

32 

4 1 

4 1 


4 -339 

6 

4.2 


— 1 


— .165 



+ 1 


— I 







8 


— 1 

— 1 



Q 

* * J 


4 -1 

4 -1 






IO 

1.4 



4 -1 

- -074 

II 

2.4 


4 -i 

. 

— .260 

12 

3-4 

. 

— I 

— I 

4 *334 


Sums 

0.000 

0.000 

0.000 

0.000 


A. 

4 -1-446 

B. 

— 0.486 

c. 

— 0.088 

D. 

4- 0.004 

Totals. 

4 1-446 

- 1.446 

- r.446 
4- 1.446 



•4 0.002 

— 0.004 
4- 0.002 

— 0.001 
4- 0.002 

— 0.001 

4 - 1-448 

- 1 -538 

-j- 0.090 

- 1-447 
4- 0.962 
4- 0.485 
4 1 -534 

— 0.960 

- 0.574 

— 0.088 

— 0.487 
4 0.575 

. 

41 

0 0 

b 0 

00 00 

00 00 ■ 

— 0.486 
-f- 0.486 

-{— 0.486 

— 0.486 



4 - 1-446 

- 1.446 

4 0.088 


00 00 

00 00 

0 0 

6 6 
l l 



— o.oco 

— 0.001 
4* 0.001 


— 0.486 
-f- 0.486 


-f- O.088 


0.000 

0.000 

0.000 

0.000 

0.000 


The first of the tables of correlates as formed above may 
now be arranged as a normal equation by application of 
formula (68) as for station adjustment (Art. 271), resulting as 
follows: 


EXAMPLE.—TABLE OF NORMAL EQUATIONS FORMED FROM 
ABOVE TABLES OF CORRELATES. 



(«) 

w 

(d 

(d) 


(Residuals.) 

a. 

4 6.000 

4 2.000 

— 2.000 

4 2.218 

— 7.890 = 

O (.OOl) 

b. 

+ 2.000 

# 

4 6.000 

4 2.000 

— 0.090 

4 0.200 

(. OOo) 

c. 

— 2.000 

4 2.000 

4 6.000 

— 1.901 

4 4.400 

(.OOO) 

d. 

4 2.218 

— 0.090 

— 1.901 

4 2.084 

- 3-427 

(. OOO) 


The symmetry of the normal equations as shown by the 
underscoring gives a partial check on their accuracy. 

281. Algebraic Solution of Normal Equations. —The 
normal equation being now arranged, it may be solved by 
elimination in tabular manner as given on page 630. 

The logarithm of each number in line 1 is. placed in 
line 2. The logarithm (= 0.77815) of the left-hand number 
is then subtracted from each of the other logarithms. The 
remainders, the logarithms of quotients, are written in line 3. 
The number corresponding to the logarithm 0.11893, in the 





































































ALGEBRAIC SOLUTION OF NORMAL EQUATION. 629 

right-hand column, is placed in a parenthesis in line 5 with a 
sign opposite to that of the number above it in line 1. 

The logarithms in line 3, columns (b), (c), and ( d), are to 
be used as the logarithms of multipliers. The sign of each 
multiplier is the opposite to that of the number above it, and 
is written in line 4. The logarithms of multipliers are next 
placed on a slip of paper, and the logarithms of products 
found by adding the logarithms of the multipliers to the 
logarithms of numbers in line 1. For example, using 
(< b ) multiplier, line 3, column ( b ), we have 

Log. of multiplier = 9.52288 sign — (Numbers); 

Log. of product ( b)(b) = 9.82391 sign — (— 0.667); 

Log. of product ( b)(c ) = 9.82391 sign -)- (+ 0.667); 

Log. of product ( b)(d ) = 9.86884 sign — (— 0.739); 

((b) by absolute term) = 0.41996 sign -j- (-f- 2.630). 

Write the numbers corresponding to these products in 
line 7, equation (b). Products belong to the equations having 
the same letter as the multiplier and in the same columns 
with the multiplicands. 

The algebraic sums of the numbers in lines 6 and 7 are 
written in line 8. In this manner form the products with the 
multipliers in (c) and (d) columns and add them to c and d 
equations respectively. 

The above process is to be repeated for the numbers in 
line 8. The products, line 16, are added to the numbers in 
line 15. Proceed as before with the numbers in line 17. 
This line has but one multiplier. The logarithm of -f- 0.995 
(line 28) is subtracted from logarithm of — 0.004; the number 
corresponding to quotient is -f- 0.004 (line 32), which is the 
value of d correlate. 

The logarithm of this number (7.60206 sign -J-) is added 
to each of the logarithms of multipliers in column (d). Thus, 
commencing at the bottom : 




630 ADJUSTMENT OF PRIMARY TRIANGULATION. 
EXAMPLE.—SOLUTION OF NORMAL EQUATIONS. 



(«) 

W 

(c) 

( d ) 

Absolute 

Terms. 

Line 

No. 

a 

4- 6.000 

-f- 2.000 

— 2.000 

4- 2.21S 

- 7.890 

I 

0.77815 

0.30103 

0.30103 

0.34596 

0.89708 

2 

0.00000 

9.52288 

9.52288 

9.56781 

0.11893 

3 

— 

— 

4 ~ 

— 

+ 

4 


(+ 0.162) 

(— 0.029) 

(— 0.002) 

( 4 - 1-315) 

5 

b 


+ 6.000 

-f 2.000 

— 0.090 

4- 0.200 

6 


— 0.667 

+ O.667 

- 0.739 

4- 2.630 

7 


4 - 5-333 

+ 2.667 

— 0.829 

4- 2.830 

8 


0.72700 

0.42600 

9-9 i8 55 

0.45179 

9 


0.00000 

9.69900 

9 - I 9 I 55 

9.72479 

IO 


— 

— 

4 ~ 


II 



(4- 0.044) 

(4- 0.001) 

(- 0.531) 

12 

c 



4- 6.000 

— 1.901 

-f 4.400 

*3 



— 0.667 

4 - 0.739 

— 2.630 

14 



4 - 5-333 

— 1.167 

4 1.770 

15 



- 1-333 

4 - 0.415 

- 1-415 

l6 



4- 4.000 

- 0.747 

4- 0.355 

17 



0 . 60206 

9*87332 

9 - 55° 2 3 

18 



0.00000 

9.27126 

8.948x7 

19 



— 

4 - 

20 




(-}- 0.001) 

(— O.089) 

21 

d 




4- 2.084 

“ 3-427 

22 




— 0.820 

4- 2.917 

23 




4- 1.264 

- 0.510 

24 




— 0.129 

4 - 0.440 

25 

Correlates. 


4 - 1.135 

— 0.070 

26 

Cl — “I - I 

.446 


— 0.140 

4- 0.066 

27 

b — — 

.486 


4- 0.995 

— 0.004 

28 

c — _ 

.088 


9.99782 

0.00000 

7.60206 
7.60424 

29 

30 


d = + 



■ 


3 T 


.004 



(4- 0.004) 

32 










































ALGEBRAIC SOLUTION OF NORMAL EQUATION. 63! 


Log. of correlate d 

added to log. in line 19, ' 

column (d), gives the log. 
of product d multiplier 
by (d) log. 

Lineio,^/multiplier by(a)log. 


= 7.60206 sign -|- 
= 6.87332 sign + No.= +.001 
= 6.79361 sign-]- No. = -f-.ooi 


Line 3, d multiplier by d log. = 7.16987 sign —, Num¬ 
ber = — .002. Each number is written in parenthesis under 
the log. of its multiplicand. 

Any line of multipliers corresponds to an equation. Take, 
for example, line 19: the numbers corresponding to each log. 
taken with the letter of the column give 


— lc + o. 187^ — 0.089 — °- 


The product of -|- 0.187 (log. 9.27126) by the value of d 
(+ .004) = .° 01 (nearest unit in third place of decimals); 

} ience — c + .° 01 — .089 = °; and 

c — — .088. 

Take the log. of c and proceed as with log. of d, obtaining 
the numbers + 0.044, line 12, and — 0.029, line 5, for products 
of c correlate by c column multipliers. 

Add algebraically the numbers on line 12 and we get the 
value of b — — 0.486. Find its product with b multiplier line 
3, and write it on line 5. Add the numbers on line 5 for the 

value of a . 

The values of d , c, b, a can also be found from lines 28, 

17. 1 - 

For example, + 4.00^ — 0.74 ?d + °-355 — 0 Oi ne 1 / ) : 
substituting the value of d (+ .004) and combining gives 

+ 4.00 c — I - 0.352 — o; and 

c = — 0.088, as before. 




632 ADJUSTMENT OF PRIMARY TRIANGULATION. 

The same operation as is illustrated on page 630 can be 
more simply and quickly performed by the method of solution 
by reciprocals and Crelle's tables , instead of by use of loga¬ 
rithms, where the former are available; see Appendix 8, 
pages 26 to 28, U. S. Coast and Geodetic Survey Report for 
1878. 

282. Substitution in Normal Equations.—The values 
found for the correlates a , b, etc., must now be substituted in 
the normal equations (Art. 280) to test the accuracy of the 
solution. 

For equation d (p. 629), commencing at the left: 

+ 2 - 2lS X (+ 1.446 = a) = + 3.207 

— 0.090 x (— 0.486 = b) = —(- 0.045 

— 1.901 x (— .088 = c) = + o. 167 

—|— 2.084 X (“I - .004 := - d') —j— 0.008 

absolute term = — 3.427 

Sum = 0.000 

This proves the equation correctly solved. In the same 

way substitute in a, b, and c equations. In equation a , as 

solved above, there is an error amounting to .001, but as only 

► 

corrections to the nearest hundredth are desired, this small 
residual may be neglected. 

283. Substitution in Table of Correlates_The values 

of the correlatives are next placed at the head of columns A, 
B, C, and D (p. 628, Table of Correlates Solved), and prod¬ 
ucts by corresponding coefficient in column a, b, c, or d> of 
the adjoining table are found and written on the proper lines 
in A, B, C, and D columns. 

The sums of the products on each horizontal line are 
placed in the column of totals. As a check on this part 
of the work see that the sum of the numbers in each col¬ 
umn = o. 

On each line in the column of totals is the correction for 



WEIGHTED OBSERVATIONS. 633 

the side adjacent to an angle, the numbers for which are given 
in the column of sides. 

For the angle 1.2.3. = (— 1.2 + 3.2) the correction is as 
follows: 

For the side 1.2 (line 4) the correction is — 1.447. 

For the side 3.2 (line 5) the correction is -j- .0962. 

Hence for the sides — 1.2 and -j- 3.2 we have 

-|- 1.447 = — correction for 1.2 

+ 0.962 = -fr correction for 3.2 

-f- 2.409 = correction for angle 1.2.3 

These are the corrections which are written in the third and 
fourth columns of the angle equation (Art. 276, p. 620) as 
side corrections and angle corrections respectively. From 
their application result the corrected spherical angles , column 
five. 

The correction for each log. sine is the product of its angle 
correction by its difference for 1" . For example, sine of 
angle 3.4.1 (Art. 279, p. 626), Dif. for 1" = — 5.9, column 
three, correction for angle = — o".66, column four. 

— 5.9 X — o.66 = -)- 3.9 = cor. to sine, column five. 

284. Weighted Observations. —Where a number of 
observations differ somewhat from each other and the causes 
are believed to be known for such difference, it occasionally 
becomes desirable to give greater value to one observation 
than to another; thus one may be given two or three times 
the value of another. This operation is called weighting 
(Art. 264). To find the weighted mean of a number of 
observations which have been given unequal weights, each is 
multiplied by its proper weight, and the sum of the product 
is divided by the sum of the weights, the quotient being the 
weighted mean. 



634 ADJUSTMENT OF PRIMARY TRIANGULATION 

Example. 38° — 54' — 55".o x Wt. 1 = 55 

54 X Wt. 1 = 54 

56 X Wt. 2 = 112 

53 X Wt. 1 = 53 

57 X Wt. 2 = 114 

Sum 7)388 

55-4 

Weighted mean = 38° 54' 55".4 

Weights are used in a least-square adjustment in the fol¬ 
lowing manner: the adjustment is carried forward as above 
described till the table of correlates is reached; then opposite 
each angle number in a station adjustment, or opposite the 
side numbers in a figure adjustment, place the weight of the 
angle or side in a separate column. Every product formed 

in the table of correlates must be divided by the weight 

written on the horizontal line with the multiplicand. The 
weight is used only as a divisor. 

The following example from a station adjustment will 
illustrate the method of using weights in station or figure 
adjustment. 

Equation a , at the bottom of page 635, is formed from 
the left half of the table on the same page, thus: 

— 1 X — 1 -f- 2 = + 0.50 

+ 1 X + 1 -f- 1 = + 1.00 

- I X — I ~2 = -f O.50 


Sum = -f- 2.00 = term (a), equation a. 
Term l?, same equation is from second line. 

+ I X — I -r- I = — I. 

Term c — o. The absolute term — 3.00 is the same that 
it would have been for an unweighted equation. 






WEIGHTED OBSERVATIONS. 


635 


Equation c is formed thus: 

Term a — o. 

Term b = -(- 1 X — 1 •+ 2 = — o. 50 

Term c = — 1 X - 1 +• 1 = + 1.0 

+ 1 X + 1 -f- 2 = -|- o. 50 

-iX-i-fi = + 3 -QQ 

Sum + 4.50 = term c. 


Example.—Table of Weighted Correlates. 


No. of 
Correc¬ 
tion. 

Weight. 

a 

b 

c 

A 

+ 2-779 

B 

+ 2.558 

C 

— 0.605 

Totals. 

Totals 

divided 

by 

Weights. 

I 

2 

— 1 



- 2.779 



- 2.779 

// 

- i 39 

2 

I 

+ 1 

— 1 

.... 

+ 2.779 

- 2.558 


-j- 0.221 

+ 0.22 

3 

2 

— 1 



- 2.779 



- 2.779 

- 1-39 

4 

I 


.... 

— I 



+ 0.605 

+ 0.605 

+ 0.60 

5 

2 


— 1 

+ 1 

...•••• 

~ 2.558 

— 0.605 

~ 3 -I 63 

- 1.58 

6 

\ 


.... 

— I 



+ 0.605 

+ O.605 

+ 1.81 

7 

4 


— 1 



- 2.558 


- 2.558 

— 0.64 

360° 

00 


+1 






0 


Example.—Equations formed from Above Correlates. 


(«) 0 ) 

a. -f- 2.000 — 1.000 

b. — I.OOO -f- 1.750 

c . — 0.500 


to w 

. — 3.00 = o 

— 0.500 — 2.00 = o 

-f- 4.500 + 4.00 = o 


The equations are solved in the ordinary way. The 
values for the correlates are multiplied by their respective 
coefficients, the products being written in columns A, B, C, 
no attention being paid to the weights until after the totals 
for each horizontal line are found and written in the column 
for totals. These totals must be divided by the weights. The 
quotients, written in the right-hand column, are the weighted 
corrections for the angles, the numbers of which are in the 
left-hand column. 











































CHAPTER XXIX. 


COMPUTATION OF DISTANCES AND COORDINATES. 

285. Geodetic Coordinates. —The position of a point on 
the surface of the earth is determined by its altitude and geo¬ 
detic latitude and longitude. These may be called its geo¬ 
graphic coordinates. In order to extend and compute a system 
of triangulation from such a point, it is necessary to determine 
also its polar coordinates , which are the distances and directions 
or azimuths between it and various other points. A complete 
statement of what are designated the geodetic coordinates of a 
point includes, therefore, its geographic and polar coordinates. 

As understood in geodetic computation the azimuth of a 
line (Art. 288) is the angle which defines its direction with 
relation to the true meridian. This angle is always measured 
from the south towards the west, north and east, in the direc¬ 
tion of the hands of a watch. The zero of azimuth is the south, 
and a true westerly direction is 90°, a northerly direction 180 0 ,. 
and an easterly direction 270°. Astronomic azimuths (Chap. 
XXXIII) and latitudes (Chap. XXXIV) are to be determined 
by observations on stars, and longitudes by the same sup¬ 
plemented by telegraphic exchanges of time (Chap. XXXV). 
Distance is obtained by direct measure (Chap. XXI) reduced 
to sea-level, of the length of the line considered. 

The computation of geodetic coordinates consists of two dis¬ 
tinct operations. The first is the computation of the lengths 
of the sides of the triangles (Art. 286) by which all the parts 
of the triangle are solved. The second operation consists in 

636 



COMPUTATION OF DISTANCES. 637 

starting out in any figure, or in the simplest figure, a triangle, 
with the lengths of the sides and dimensions of the angles 
known, as well as with the latitude and longitude of one, or, 
preferably, for purposes of check, of two apices of the triangle 
known. Also, the azimuth of known side joining the two 
known positions. With these quantities given it is possible 
to compute the latitudes and longitudes of the other apices or 
stations and the azimuths of the lines joining them (Art. 288). 

286. Computation of Distances. —The figure adjustment 
having been completed and the spherical excess in each 
triangle computed (Arts. 273 and 277), the next operation is 
the computation of the distances or lengths of each of the 
sides forming the various triangles. In each triangle there 
is a known base, that is, the length of one side is known and 
the three angles are known. The remaining sides are com¬ 
puted on the principle of proportion of sides to sines of opposite 
angles ; expressed mathematically this is 


b sine A 
sine B 



In this computation the distances are expressed in logs, of 
meters because the tables used in the after-computation are 
prepared for metric computations. The solution of the above 
formula is best explained graphically by the following example 
(see also Fig. 178). 


Example. 


Triangle. 


Stations. 


Spherical Angles. 


£•«• 


Plane Angles. 


Log. Sines. 


Log. sine angle at McKenzie... 9.741 7780 
Log. distance Chuska—Zuni... 4.501 6173 



McKenzie 

33 ° 

29' 

26".37 

- 1.42 

33 ° 

29' 

24 // -95 

0.258 2220 a. c. 

I 

Chuska 

73 

15 

40 -43 

- 1.42 

73 

15 

39 -oi 

9.981 1959 


Zuni 

73 


57 -46 

- 1.42 

73 

14 

56 .04 

9.981 1686 


180 00 04.26 4.26 

Log. side McKenzie—Chuska 4.741 0079 

Log. side McKenzie—Zuni. 4.7410352 


















638 COMPUTATION OF DISTANCES AND COORDINATES. 

In the above, under the column of stations, is placed first 
the station from which the distance is to be determined, and 
then follow those stations to which distances are to be deter¬ 
mined and between which the distances are already known. 
In the column of spherical angles are written the final adjusted 
angles resulting from the figure adjustment (Art. 276). In 
the column of spherical excess is written opposite each angle 
one-third of the total spherical excess of the triangle (Art. 
277). In column of plane angles are written the angles 
resulting from the subtraction of spherical excess from the 
adjusted spherical angles. 

In the column of log. sines are written logarithmic sines of 
the plane angles as obtained from a table of logarithms. 
Above this column is written opposite “log. of dist.” the 
length of the known side Chuska-Zuni, and above it, for 
reference, the log. sine of the angle at the known station, 
McKenzie. On the line McKenzie, in column of log. sines, is 
then written the arithmetical complement, a. c., of the sine 
of its angle. Opposite the other two angles, Chuska and 
Zuni, are written the logarithms of their sines. 

Th & quantities sought, viz., distances McKenzie—Zuni and 
McKenzie—Chuska, are found by adding the a. c. log. sine of 
the angle at McKenzie and the log. sine of the angle at Chuska 
in the first case, and in the second the a. c. log. sine of the 
angle at McKenzie and the log. sine angle at Zuni to the 
log. dist. Chuska—Zuni. 

287. Formulas for Computing Geodetic Coordinates. 

The last operation in the computation of a system of primary 
triangulation is the determination of the geodetic coordinates 
of each station. Having now the log. dist. and accordingly 
the actual lengths of the sides and the dimensions of each 
plane angle, there remains only to determine the latitude and 
longitude of each station and the azimuth of each direction. 
To do this the latitude and longitude of one station must be 
known, and the azimuth from it to one of the other stations. 


COMPUTATION OF GEODETIC DISTANCES. 639 


The formulas for computing new latitudes, longitudes, 
and azimuth from the known positions consist in deter¬ 
mining the differences of latitude, longitude, and azimuth, 
Acf), AX, and Aa, and adding or subtracting these to or from 
the known positions. 

— A(p = S cos a.B -|- S* sin 3 a. C 4 -(A(/>)*. D — /*.sin 3 a . E. (73) 

The above is simple of application by use of the log. factors 
B, C, D, E y etc., Table XXXVII. 


5 sin a 

AX= - —j A' t . 

COS 0 

sin £(0+ 0') 


(74) 


— Aa — AX 


cos i{A(p) 


4 ~ AX*F t . . (75) 


and 


a' = o'-}- 180 0 -}■ Aa. 


(7 6) 


The constants from Table XXXVII have the following 
algebraic values, and the notation used is as given below: 


B = 


C = 


1 


R arc 1 


it 


tan 0 


2NR arc 1 


n 


(1 - e % sin* 0)* 
a(i — « a ) arc 1 " ’ 

(1 — e* sin 2 0) 3 tan 0 
2 a\i — e 2 ) arc 1" 


• • • • (77) 


; . . . . (78) 


D = 


|e sin 0 cos 0 arc 1 
(1 — e 1 sin 2 0)- 


• • 


• • (7 9 ) 


E = 1 + 3 tan 2 0 _ (14-3 tan 3 0)( 1 - e* sin 2 0) # ^ 


6N 


6 a 


A' = 


1 


(1 — e* sin 2 (p)\ 


N arc 1 


n 


a arc 1 


rr 


(referred to new position;) (8 I) 















64O COMPUTATION OF DISTANCES AND COORD 1 NATES. 
in which 

N — ---, = normal ending at minor axis; 

(1 — e 1 sin 3 0)- 

a = equatorial radius (Art. 292); 

N 3 

R — —(1 — ej = radius of curvature of the meridian; 
a 1 

p = N cos 0 = radius of curvature of the parallel; 
e = eccentricity (Art. 292); and 


log F — sin 0 cos’ 0 arc' 1',. . . (82) 


h = sB cos a or first term. 

Also 0 = latitude of known station, + if north, 

X — longitude of known station, if west; 

oc = azimuth to second station from first, with 
careful regard of algebraic signs, 

0', X\ and a' = same for new or required position. 


For distances less than twenty-jive miles omit the quantities 
depending on the constants D and A, which give the logs. 
(Ill) and (IV) in example on page 644. Also omit the small 
log. cor. to AX depending on log. (V) ; the small correction to 
A a depending on log. F\ and that to log. (VI) derived from 


log. sec. 




For distances greater than one hundred miles the following 
formulas should be employed ; 


tan %(<*' + C — AX) 


sin i(y — 0) a 
sin i(y -f 6 ) Cot 2 



tan £(«' + C + A A) 


cos j(y — 6 ) 

c©s i(y -f 6 ) cot 2 ’ 


(84) 







FORMULAS . 


641 


0 '- 0 =- 


siri'2^6t' —J— C — oLj 


psin 1 sin^(a' —(— Q— J— cd) 


^ a sin a i" "I 

_H- ^ 2 — C0S ’i(«-«)J : ( 8 5) 


in which y 
<p m 


P = 


6 - 


colatitude of old point; 
mean latitude of old and new points, 
«(i — o 


(1 — e 2 sin' 0“) 
s 




r sin 1 


// * 


a 


c = 


(1 — e 2 sin a 0)* 
e'P sin 1" 


4(1-0 


Cos a 0 sin 2a ; 


f 

which are constants for 0ie particular case. 

In terms of the coordinates of rectangular axes referred 
to one of the points of the triangulation, the latitude and 
longitude of which are known ,—y being the ordinate in the 
direction of the meridian, and ^ the ordinate perpendicular 
to it,—the values may be expressed: 

+' =<p± r£v~ isin l iw£v) xtan ( 0 ± SF') : ( 86 > 


X' = A d: 


x 


\N sin 1 


7, x 


COS 0 ' 


(87) 


a'=- (18o° -)- a) ± 


iVsin 1 


t, tan 0'. 


( 88 ) 


The convergence of meridians, or the amount by which the 
azimuth at one end of a line exceeds the azimuth at the other, 
is expressed by the quantity 


U — --^sin i (0 + 0 ') or (A' — A) sin J (0 -f- 0'). . 

COS 0 


(89) 


















642 COMPUTATION OF DISTANCES AND COORDINATES. 

288. Computation of Geodetic Coordinates : Example. 

—From Fig. 178, platted roughly in proper relation to the 
points of the compass, we are able to ascertain the mode of 
determining the azimuths between the various stations. The 
known side, Chuska—Zufii, is drawn heavier than the others, 
and in the following computations the geodetic coordinates of 

N 


MC KENZIE 

Fig. 178.—Computation of Azimuths. 

both Chuska and Zufii are supposed to have been determined 
previously, the coordinates of McKenzie being desired. 
Drawing north and south lines through Chuska and Zufii, it 
is evident that to obtain the azimuth Chuska—McKenzie 360° 
must be added to the azimuth Chuska—Zufii and the spherical 
angle at Chuska be subtracted from this, the result being the 
azimuth Chuska—McKenzie. Likewise, knowing the azimuth 

of the line Zufii—Chuska, the azimuth of the line Zufii_ 

McKenzie is obtained by adding to the former the spherical 
angle at Zufii. 

On pages 644 and 645 is an example of the method of 
computing geodetic coordinates. The order of computation 
consists— 

First, in determining the new azimuths , as just described, 
and as illustrated at the top of the pages of example. 




COMPUTATION OF GEODETIC COORDINATES. 643 


Second, the latitude of the new point is obtained as shown 
in the left-hand column of either page. The latitude of the 
known point, as Chuska, is written opposite 0 , then a differ¬ 
ence of latitude, ^0, is obtained through the process of the 
entire computation shown in the left-hand columns. This 
amount, which is found at the bottom of the left side of the 
page, is then written under the latitude of Chuska with its 
proper algebraic sign, and subtracting it, in this case, the 
latitude of the unknown point, McKenzie, is found. 

Third, the longitude of the new point is obtained as shown 
in the columns on the right-hand side of the pages. These 
are arranged in manner similar to the latitude computation by 
writing the longitude of the known point, A, and then deter¬ 
mining a difference of longitude, AX f which in the example is 
minus and is therefore subtracted from the longitude of 
Chuska to obtain the longitude of the unknown point, 
McKenzie. The signs in both of the above cases can be veri¬ 
fied by a diagram showing the relative positions of the points. 
Such a diagram (Fig. 178) shows clearly that McKenzie is 
south of Chuska and its latitude less, and that it is also east 
of Chuska and its longitude therefore less. 

Fourth, the azimuth computation is performed as shown 
in the lower part of the right-hand columns. This consists in 
determining the back or the reverse azimuth from McKenzie 
to Chuska or Zuni. This is done by determining the differ¬ 
ence of azimuths, Aa, which is written at the top of the right- 
hand column with its proper sign. The latter can be verified 
again by reference to the diagrammatic figure. 

Finally, at the extreme bottom of the right-hand column 
is a test of the azimuth computation. This check is had by 
subtracting the back azimuths one from the other and noting 
if the result is equal to the spherical angle at McKenzie. 

Latitude , longitude , and azimuth , or 0 , A, and <*, of the 
known points, must have been previously determined by 
astronomic observations (Chaps. XXXIII to XXXV) or from 


644 COMP UTA TION OF DISTANCES AND COORD IN A TES. 


Azimuth a : 

Chuska — 

Zuhi 

65° 

22 ' 

46".27 

Spherical Angle : 

at Chitska 


-73 

15 

40 .43 

Azimuth a': 

Chuska — 

McKenzie 

352 

07 

05 .84 

An + i8o° 



ISO 

02 

37 .71 

Azimuth ( a): 

McKenzie — 

Chuska 

172 

09 

43 .55 




Geodetic Coordinates. 



LATITUDE. 



LONGITUDE. 

0: 33° IT 

Atp — 29 

32". 00 

31 .25 

Chuska 

X: 

AX: 

109 0 

57 ' 09".23 
-4 49 .98 

0 32 43 

00 .75 

McKenzie 

X’ 

109 

52 19 .25 

Computation for latitude 

* 

• 

Computation for longitude 

log. s 
“ B 

** cos a' 

4.7410079 

8.5113533 

9.9958779 

Table XXXVIII 

cor. for log. s — — 55 
cor. for AX = + 01 

log. s 
“ sin cd 
“ A' 

“ sec. 0' 

4.7410079 

9.1371272 

8.5092972 

0.0749413 

log. (I) 

3.2482391 

log. cor. — — 54 

“ (V) 


2.4623736 

log. s 2 

4 ‘ C 
'** sin 2 cd 

i 

9.48201 

1.22086 

8.27425 


AX = 

- 289 

- 54 
682 

".980 


log (II) 

8.97712 


Computation of azimuth .* 





log. (V) 


2.462368 

log. D 

2.3541 

0 + <P 


,« • / 

'04-0'\ 


“ [H-ii] 

1 6.4965 

2 

32 56 401.38 

sin.f 

7 ) 

9.735480 

log. (Ill) 

8.8506 



** sec.j 

;?) 

0.000004 

log. E 

5.9702 



log (VI) 


2.197852 

“ s 2 sin 2 < 

a' 7.7562 



— Aa 


157". 71+ 

“ (I) 

3.2482 



— 


24 37" .71+ 

ilog. (IV) 

6.9740 






^(1) 177T 

.084~ f* 






(II) 

.095+ 






< (HI) 

.071+ 

[I + II] 

1771". 18 




(IV) 

.001- 

log. “ 

3 .248263 





“ [I + II] 1 


— 2/0 mi".2 19 
— — 29' 31".25 


6 .496526 






















COMPUTA TION OF GEODETIC COORD IN A TES. 645 


Azimuth a: 

Zuhi 

— Chuska 

245 0 

13' 

3 8".2 4 

Spherical Angle: 


at Zuhi 

73 

14 

57 .45 

Azimuth a’-. 

Zuhi 

— McKenzie 

318 

27 

35 .69 

A a + 180 0 * 



180 

12 

41 .49 

Azimuth (a) 

McKenzie 

— Zuhi 

138 

40 

17 .18 


Geodetic Coordinates. 


LATITUDE. 


LONGITUDE. 


<t>: 

Acp 

33 0 

of 

22 

21". 37 

20 .62 

Zuhi 

A: 

AX 

110° 

15' 41".67 

23 22 .42 

<P 

32 

42 

00 .75 

McKenzie 

X' 

109 

52 19 .25 


Computation for latitude 

• 

• 

Computation for longitude ,* 

log. s 

4.7410352 

Table XXXVIII 

log. s 

4.7410352 

“ B 

8.5113618 


“ sin. a' 

9.8216079 

“ cos a! 

9.8741872 

cor. for log. s = — 55 

“ A' 

8.5092972 



cor. for log. AX = + 34 

“ sec. (p r 

0.0749413 

log. (I) 

3.1265842 






log . cor. — — 21 

log. (V) 

3.1468816 

log. s 2 

9.48207 


- 21 

“ C 

1.21890 



795 

“ sin. o' 

9.64321 


AX = - 1402 ". 424 


\ 


log. (II) 0.3 44 18 


Computation of azimuth .* 


log. D 2.3532 

“ [I-i-II ] 2 6.2546 


log. (Ill) 8.6078 


log. (V) 3.146880 

j 9.734780 

" sec. 0.000002 


= 32° 5 ^ 22".06 " sin. 

2 


log. E 5.9679 

“ s 2 sin 2 or' 9.2253 
(i) 3.2255 


log. (IV) 3.2293 


log. (VI) 2.881662 

— 761". 49+ 

= 22* 42". 49+ 


Azimuth check. 


(I) 1338".592+ 

(II) 2 .209f 

(III) M40+ 

(IV) .017- 


— A $13 40 ". 624 + 
= - 22 ' 20 ". 624 


McK. — Chus 172 0 09' 48 ".55 
__ “ — Zuhi 138 40 17 .18 

[I4-II] 1340.601 - 

log. “ 3.127299 Check: 33 29 26.37 

Spher. angle 

“ [1-HIP 6.254598 at McKenzie 33 29 26 .37 
























646 COMP U TA TION OF D IS TA NCES A ND CO ORD IN A 7 ES. 

computation; the log. or length of the known side, has 
been obtained from previous computation (Art. 286). The 
quantities B , C, D, and E are obtained from Table XXXVII 
by using the known latitude, 0 , as an argument. The quan¬ 
tity A' comes from the same table by using the new latitude, 
0', as an argument. 

The small correction dif. log. ^ is obtained from Table 
XXXVIII by using log. ^ as an argument. The cor. AX is 
obtained from the same table by using the argument log. (V), 
which is log. AX. In applying these corrections signs must 
be carefully watched. The resulting log. cor. is applied, with 
attention to signs, as a correction to log. (V). 

The log. F, which is added to cor. AX\ is obtained from 
Table XL, using latitude 0 as an argument. This correction 
is very small and is only made in cases of very long distances, 
as in the illustration used. 

The sec. of 0 ' = a. c. log. cos 0 ' = - - 

cosin 0 

A(p 

The sec. — is obtained from Table XXXIX. 

2 

289. Knowing Latitudes and Longitudes of Two 
Points, to Compute Azimuths and Distances. —This is the 
inverse problem of that considered in the preceding article. 
It not infrequently occurs when the latitudes and longitudes 
of two positions are known that it is desired to find the dis¬ 
tance between them and their mutual azimuths. This problem 
may prove useful in exploratory surveying when the latitudes 
of two intervisible mountain peaks which differ but little in 
longitude can be determined by sextant or transit, and longi¬ 
tudes by flashing signals or by chronometer. Then by this 
problem a base for triangulation may be obtained which will 
enable the explorer to rapidly extend the area of his survey. 
Providing the initial points are at a considerable distance 




INVERSE PROBLEM . 


64; 


apart, the station error will be inappreciable for small-scale 
mapping. Again, in a system of triangulation it may be 
desired to ascertain whether two stations hidden by trees, 
haze, etc., are intervisible. Providing their latitudes and 
longitudes are known or can be computed from other triangula¬ 
tion points, their initial azimuths may be found from this 
problem, when it will be possible to set up an instrument at 
one station and lay off the exact azimuth to the other for 
guidance in clearing timber or in heliotroping. 

This problem can be readily solved in tabular manner by 
arranging the form of solution used on pp. 644 and 645. 

To do this divide AX = sin a . A sec 0 ' by the first term 
for A(p, h = s cos a . B, whence we get 


tan a — 


AX.B 
A sec 07 / 


(90) 


If h were known, this would give the azimuth at once, 
since AX is given. 

The following example shows the method of performing 
the operation. The northernmost point should be used as 
the initial position, then all signs for (I), (II), and (III) are 

and for (IV) —. The value of AX may be either -f- or —, 
but this sign need only be used in determining in which 
quadrant the azimuth angle oc falls, i.e., the sign of tan a 
(12). An inspection of a rough plot of the position will 
also determine this. The correction to AX is found from a 
distance scaled off from the plot, and need not be very 
close. In (8) the term (I + If) 2 is the square of the difference 
of latitude Acp in seconds. Since (IV) is always small, 
log (I) in ( 0 ) may be taken as log of AX from (10). If 
cos a is smaller than sin a, find j from log ^ cos a in (11). 
As a check on the work compute the second position, using 
distance and azimuth found as above. The order of solution 
is shown by figures in parentheses. The cosines of latitudes 
are proportional to the intercepted parallels. 




648 COMPUTATION OF DISTANCES AND COORDINATES . 


The results obtained from this problem should be checked 
by computing latitudes and longitudes by the direct method 
as shown in the example on pages 644 and 645. 


Latitude. 

(p = 38° 23' 27".O 
0 = 37 45 09 .3 


Longitude. 

104° 32' 48".2 = A 
104 49 05 .5 = A' 


A<p — 38' 17".7 — (p — (p' AX — 16' 17".3 -f- 

= 2297".7 (1) = 977"-3 + 

log A(p — 3.3612933 

log AX = 2.9900279 -f- 

log C 1.30360 correction to A A 83 -f- 

log s* sin 5 a 8.75770 - 

- AX' 2.9900362 


0.06130 (7) 

(II) = 1 ".152 cor. AX 17 + 

As 100 — 


(2) 


( 4 ) 


log D 2.3812 

log (I -f II) S 6.7226 


83 ~ ( 3 ) 

log A' = 8.5091750 (5) 


9.1038 (8) 


sec cp' — o. 1020092 


(HI) = +o".i 3 


8.6111842 (—) 
log AX' = 2.9900362 (-)-) 


log E 6.0711 
log j 2 sin 2 a 8.7577 (9) 

log (I) 3-3613 

8.1901 

(IV) = - 0."02 

(II) = + V'.is 

(III) = + -13 

(IV) = — .02 (10) 


Sum = + i".26 

A<p 2297.7 


log j sin a = 4.3788520 (4-) (6) 

log j cos a = 4.8500742 ( —) 


sin a 
cos a 


= tan a = 9.5287778 


log (I) = 3-3610475 
log B — 8.5109733 


(12) 


log S COS (X = 4.8500742 (11) 

Azimuth = a— 18 0 40' io".8 (13) 

(or 180 -}- this angle) 

log s sin a = 4.3788520 
log sin « = 9-5053013 (14) 


(I) = 2296".4 = difference Distance (log) = log s = 4.8735507 
















FACTORS USED IN GEODETIC COMPUTATIONS. 649 

Table XXXVII. 

FACTORS FOR COMPUTATION OF GEODETIC LATITUDES, 
LONGITUDES, AND AZIMUTHS. 

(From Appendix No. 9, Report of U. S. Coast and Geodetic Survey, 1894.) 

LATITUDE 30 °. 


Lat. 

log A 

diff. \" — — o.c6 

log B 

diff. 1 " = — 0.19 

log C 

diff. — +0.48 

log D 

diff. 1" = -f" 0.02 

log E 

diff. 1" = -f- 0.05 

O / 

30 OO 

8.509 3588 

8.511 5729 

1.16692 

2.3298 

59127 

I 

84 

18 

721 

99 

3° 

2 

81 

06 

75o 

2.3301 

33 

3 

77 

8.511 5695 

778 

02 

36 

4 

73 

84 

807 

04 

39 

05 

69 

73 

836 

05 

42 

6 

66 

62 

865 

06 

45 

7 

62 

5 X 

894 

08 

48 

8 

58 

40 

923 

09 

5 X 

9 

55 

28 

952 

II 

54 

IO 

8.509 3551 

8.511 5617 

1.16981 

2.3312 

5-9157 

II 

47 

06 

I,17010 

x 4 

59 

12 

43 

8-5 11 5595 

039 

*5 

62 

13 

40 

84 

068 

*7 

65 

M 

3 6 

73 

097 

18 

68 

15 

32 

6l 

126 

1 9 

7i 

l6 

29 

50 

i55 

21 

74 

17 

25 

39 

184 

22 

77 

18 

21 

28 

212 

24 

80 

X 9 

17 

17 

241 

25 

83 

20 

8.509 35 x 4 

8*5” 5505 

x.17270 

23327 

5.9x86 

21 

JO 

8.5” 5494 

299 

28 

89 

22 

06 

83 

328 

30 

92 

23 

02 

72 

357 

3 1 

95 

24 

8.509 3499 

6l 

385 

32 

98 

25 

95 

49 

414 

34 

5.9200 

26 

9 1 

38 

443 

35 

03 

27 

88 

27 

472 

37 

06 

28 

84 

l6 

500 

38 

09 

29 

80 

04 

529 

39 

12 

3° 

8.509 3476 

8.5 11 5393 

x .i7558 

2 *334 T 

5-9215 

3 1 

72 

82 

587 

42 

18 

32 

69 

7i 

615 

44 

21 

33 

65 

59 

644 

45 

24 

34 

6l 

48 

673 

47 

27 

35 

57 

37 

701 

48 

3° 

36 

54 

26 

730 

49 

33 

37 

50 

r 4 

759 

5 x 

36 

38 

46 

03 

788 

52 

39 

39 

42 

8.511 5292 

816 

54 

42 

40 

8.509 3439 

8.511 5281 

1-17845 

2-3355 

5.9245 

41 

35 

69 

874 

56 

48 

42 

3 1 

58 

902 

58 

5 1 

43 

27 

47 

93 1 

59 

53 

44 

24 

35 

959 

60 

56 

45 

20 

24 

988 

62 

59 

46 

l6 

*3 

1.18017 

63 

62 

47 

12 

02 

045 

65 

65 

48 

09 

8.511 5190 

' 074 

66 

68 

49 

05 

79 

102 

67 

7 1 

5° 

8.509 3401 

8.511 5168 

1.18131 

2.3368 

5.9274 

51 

8-509 3397 

56 

160 

70 

77 

52 

94 

45 

188 

7 1 

80 

53 

90 

34 

217 

73 

83 

54 

86 

22 

245 

74 

86 

55 

82 

I ( 

274 

76 

89 

56 

78 

OO 

302 

77 

92 

57 

75 

8.511 5088 

33 l 

78 

95 

58 

7 1 

77 

359 

80 

98 

59 

67 

66 

388 

81 

5.9301 

60 

8.509 3363 

8.511 5054 

1.18416 

2.3382 

5-9304 

- 



















£> 5 ° COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVII. 


FACTORS USED IN GEODETIC COMPUTATIONS. 

LATITUDE 31 °. 


Lat. 

log A 

log B 

log C 

log D 

log E 

diff. \ n = — 0.06 

1 

diff. = — 0.19 

diff. =■ 0.47 

diff. 1" = -f- o.oe 

diff. 1" — -{-0.05 

o f 

3 1 00 

8.509 3363 

8.511 5054 

1.18416 

2.3382 

5-9304 

I 

60 

43 

445 

84 

07 

2 

56 

32 

473 

85 

IO 

3 

52 

20 

501 

86 

*3 

4 

48 

09 

530 

88 

16 

05 

44 

8.511 4998 

558 

89 

r 9 

6 

41 

86 

587 

90 

22 

7 

37 

75 

615 

92 

25 

8 

33 

64 

643 

93 

28 

9 

29 

52 

672 

95 

3 1 

IO 

8.509 3325 

8.511 4941 

1.18700 

2.3396 

5-9334 

II 

22 

29 

729 

97 

37 

12 

18 

18 

757 

99 

39 

13 

14 

07 

785 

2.3400 

42 

14 

10 

8.511 4895 

813 

OI 

45 

15 

06 

84 

842 

02 

48 

l6 

03 

72 

870 

04 

5* 

17 

8.509 3299. 

6l 

898 

05 

54 

18 

95 

50 

927 

06 

57 

19 

91 

38 

955 

08 

60 

20 

8.509 3287 

8.511 4827 

1.18983 

2.3409 

5.93 6 3 

21 

84 

x 5 

1.19012 

IO 

66 

22 

80 

04 

040 

12 

69 

23 

76 

8.511 4793 

068 

1 3 

' 72 

24 

72 

8l 

096 

*4 

75 

25 

68 

70 

125 

l6 

78 

26 

65 

58 

x 53 

*7 

81 

27 

61 

47 

l8l 

18 

84 

28 

57 

35 

209 

20 

87 

29 

53 

24 

238 

21 

90 

30 

8.509 3249 

8.511 4713 

1.19266 

2.3422 

5-9393 

31 

46 

OI 

294 

23 

96 

32 

42 

8.511 4690 

322 

25 

99 

33 

38 

78 

351 

26 

5-9402 

34 

34 

67 

379 

27 

05 

35 

30 

55 

407 

29 

08 

36 

26 

44 

435 

30 

II 

37 

23 

32 

463 

3 1 

x 4 

38 

19 

21 

491 

33 

l 7 

39 

x 5 

09 

520 

34 

20 

40 

8.509 3211 

8.511 4598 

1.19548 

2-3435 

5.9423 

41 

07 

86 

576 

36 

26 

42 

03 

75 

604 

38 

29 

43 

OO 

63 

632 

39 

32 

44 

8-509 3 x 96 

52 

660 

40 

35 

45 

92 

40 

688 

4 1 

38 

46 

88 

29 

716 

43 

41 

47 

84 

x 7 

744 

44 

44 

48 

8 t 

06 

772 

45 

47 

49 

77 

8.511 4494 

800 

47 

50 

5° 

51 

8.509 3 X 73 

69 

8-5 11 4483 

7 i 

1.19828 

856 

2.3448 

49 

5-9453 

S 6 

52 

65 

60 

884 

50 

59 

53 

61 

48 

912 

52 

62 

54 

57 

37 

940 

53 

65 

55 

54 

25 

968 

54 

68 

56 

50 

x 4 

996 

55 

72 

57 

46 

02 

1.20024 

57 

75 

58 

42 

8.511 4391 

052 

58 

78 

59 

38 

79 

080 

59 

81 

60 

8.509 3134 

8.511 4368 

1 .20108 

2.3460 

5.9484 






















FACTORS USED IN GEODETIC COMPUTATIONS . 6$ I 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 32 °. 


Lat. 

log .4 

diff. 1" = — 0.06 

log B 

diff. 1" 0.19 

log C 

diff. 1" = -|-o.46 

log D 

diff. = -j-0.02 

log £ 

diff. 1" = -(-0.05 

O / 

32 OO 

8.509 3i34 

8.511 4368 

1.20108 

2.3460 

5.9484 

I 

31 

56 


62 

87 

2 

27 

44 

164 

63 

90 

3 

23 

33 

192 

64 

93 

4 

19 

21 

220 

65 

96 

05 

15 

IO 

248 

67 

99 

6 

I I 

8.511 4298 

276 

68 

5.9502 

7 

07 

87 

304 

69 

05 

8 

04 

75 

332 

70 

08 

9 

OO 

63 

360 

7« 

II 

TO 

8.509 3096 

8.511 4252 

1.20387 

2-3473 

5-95*4 

I I 

92 

40 

415 

74 

17 

12 

88 

29 

443 

75 

20 

'3 

84 

J 7 

47 l 

76 

23 

14 

SO 

05 

499 

78 

26 

15 

76 

8.511 4194 

527 

79 

29 

16 

73 

82 

555 

80 

32 

17 

67 

7i 

582 

81 

35 

18 

65 

59 

610 

82 

38 

19 

6l 

47 

638 

84 

41 

20 

8.509 3057 

8.511 4136 

1.20666 

2.3485 

5-9544 

21 

53 

24 

694 

86 

47 

22 

49 

13 

722 

87 

50 

23 

46 

OI 

749 

88 

53 

24 

42 

8.511 4089 

777 

90 

56 

25 

38 

78 

805 

9i 

60 

26 

34 

66 

833 

92 

63 

27 

30 

54 

860 

93 

66 

28 

26 

43 

888 

94 

69 

29 

22 

3i 

916 

96 

72 

3° 

8.509 3018 

8.511 4020 

1.2C944 

2.3497 

5-9575 

3 1 

15 

08 

971 

98 

78 

32 

11 

8.511 3996 

999 

99 

81 

33 

07 

85 

1.21027 

2.3500 

84 

34 

03 

73 

054 

02 

87 

35 

8.509 2999 

6l 

082 

03 

9o 

36 

95 

50 

no 

04 

93 

37 

91 

38 

137 

05 

96 

38 

87 

26 

*65 

06 

99 

39 

83 

15 

193 

07 

5.9602 

4° 

8.509 2980 

8.511 3903 

I.21220 

2-3509 

5>96o5 

41 

76 

8.511 3891 

248 

IO 

08 

42 

72 

79 

276 

1 1 

I I 

43 

68 

68 

3°3 

12 

15 

44 

64 

56 

33 1 

*3 

18 

4 ^ 

60 

44 

358 

14 

21 

46 

56 

33 

386 

l6 

24 

47 

52 

21 

414 

17 

27 

48 

48 

09 

44' 

15 

30 

49 

44 

8.511 3798 

469 

19 

33 

5° 

8.509 2940 

8.511 3786 

x.21496 

2.3520 

5.9636 

51 

37 

74 

524 

21 

39 

52 

33 

63 

55 1 

23 

42 


29 

51 

579 

24 

45 

54 

25 

39 

607 

25 

48 

CC 

21 

27 

634 

26 

51 

% X 

17 

l6 

662 

27 

54 

S7 

!3 

04 

689 

28 

58 

58 

09 

8.511 3692 

717 

29 

6l 

59 

05 

80 

744 

3 r 

64 

to 

8.509 2901 

8.511 3669 

1.21772 

2-3532 

5.9667 
















652 COMPUTATION OF DISTANCES AND COORDINATES . 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 33 ° 


r- 

Lat. 

I log ^ 

1 d iff. 1" = — 0.07 

log B 

diff. \" — — 0.20 

log C 

diff. 1" = +o -45 

log D 

diff. 1"= + 0.02 

| log E 

diff. -f - 0.05 

O / 

33 00 

8.509 2901 

8.511 3669 

I.21772 

2.3532 

5.9667 

I 

8.509 2897 

57 

799 

33 

70 

2 

94 

45 

827 

34 

73 

3 

90 

33 

854 

35 

76 

4 

86 

22 

882 

36 

79 

05 

82 

IO 

909 

37 

82 

6 

78 

8.5H 3598 

937 

38 

85 

7 

74 

86 

964 

40 

88 

8 

70 

75 

992 

4 i 

92 

9 

66 

63 

I.22019 

42 

95 

IO 

8.509 2862 

8 - 5 ” 355 i 

1.22047 

2-3543 

5.9698 

II 

58 

39 

074 

44 

5.9701 

12 

54 

28 

101 

45 

04 

13 

51 

l6 

129 

46 

07 

M 

47 

04 

156 

47 

10 

15 

43 

8.511 3492 

184 

49 

13 

l6 

39 

80 

211 

50 

l6 

17 

35 

69 

238 

5 i 

*9 

18 

3 » 

57 

266 

52 

22 

19 

27 

45 

293 

53 

26 

20 

8.509 2823 

8- 5 ” 3433 

I .22321 

2-3554 

5-9729 

21 

19 

21 

348 

55 

32 

22 

15 

IO 

375 

56 

35 

23 

11 

8.511 3398 

4°3 

57 

38 

24 

07 

86 

43 ° 

58 

41 

25 

03 

74 

457 

60 

44 

26 

8.509 2799 

62 

485 

6l 

47 

27 

95 

5 i 

512 

62 

50 

28 

91 

39 

539 

63 

53 

29 

88 

27 

567 

64 

57 

30 

8.509 2784 

8 - 5 ” 33 T 5 

1.22594 

2.3565 

5.9760 

31 

80 

03 

621 

66 

63 

32 

76 

8.511 3291 

648 

67 

66 

33 

72 

80 

676 

68 

69 

34 

68 

68 

703 

69 

72 

35 

64 

56 

73 ° 

70 

75 

36 

60 

44 

757 

71 

78 

37 

56 

32 

785 

73 

81 

38 

52 

20 

812 

74 

85 

39 

48 

09 

839 

75 

88 

40 

8.509 2744 

8.511 3197 

1.22866 

2-3576 

5 . 979 1 

41 

40 

85 

893 

77 

94 

42 

36 

73 

921 

78 

97 

43 

32 

6l 

948 

79 

5.9800 

44 

28 

49 

975 

80 

03 

45 

24 

37 

1.23002 

81 

06 

46 

20 

25 

029 

82 

IO 

47 

l6 

1.3 

057 

83 

13 

48 

12 

02 

084 

84 

16 

49 

08 

8.511 3090 

III 

85 

J 9 . 

50 

8.509 2704 

8.511 3078 

1.23138 

2.3586 

5.9822 

5 i 

OI 

66 

165 

87 

25 

52 

8.509 2697 

54 

192 

88 

28 

53 

93 

42 

220 

89 

31 

54 

89 

30 

247 

9 i 

35 

55 

85 

18 

274 

92 

38 

56 

81 

06 

3 ° I 

93 

41 

57 

77 

8.511 1995 

328 

94 

44 

58 

73 

83 

355 

95 

47 

59 

69 

71 

382 

96 

50 

60 

8.509 2665 

8 - 5 ” 2959 

1.23409 

2-3597 

5.9853 



















FACTORS USED IN GEODETIC COMPUTATIONS. 653 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 34 ° 


Lat. 

log A 

diff. ' — — 0.07 

log B 

diff. 1" =-0.20 

log C 

diff. 1"= -f 0 45 

log D 

diff. l''^: -|- 0.02 

log E 

diff. 1"= -f- 0.05 

O / 

34 OO 

I 

8.5O9 2665 

6l 

8.511 2959 

47 

1.23409 

437 

2-3597 

98 

5-9853 

57 

2 

57 

35 

464 

99 

60 

3 

53 

23 

49 1 

2.3600 

63 

4 

49 

11 

518 

OI 

66 

05 

45 

8.511 2899 

545 

02 

69 

6 

41 

87 

572 

03 

72 

? 

37 

75 

599 

04 

75 

8 

33 

63 

626 

05 

79 

9 

30 

5i 

653 

06 

82 

IO 

8.509 2625 

8.511 2840 

1.23680 

2.3607 

5-9885 

1 I 

21 

28 

707 

08 

88 

12 

17 

l6 

734 

09 

9 1 

*3 

*3 

04 

761 

IO 

94 

14 

09 

8.511 2792 

788 

I I 

97 

*5 

*°5 

80 

815 

12 

5.9901 

l6 


68 

842 

13 

04 

17 

8.509 2597 

56 

869 

14 

07 

18 

Q 3 

44 

896 

*5 

IO 

19 

89 

32 

923 

16 

13 

20 

8.509 2585 

8.511 2720 

1.23950 

2.3617 

5.99 16 

21 

8l 

08 

977 

18 

19 

22 

77 

8.511 2696 

I.24OO4 

J 9 

23 

23 

73 

84 

031 

20 

26 

2 4 

69 

72 

058 

21 

29 

25 

65 

60 

085 

22 

32 

26 

6l 

48 

112 

a 3 

35 

27 

57 

36 

«39 

24 

38 

28 

53 

24 

165 

25 

42 

29 

49 

12 

192 

26 

45 

3° 

8.509 2545 

8.5112600 

1.24219 

2.3627 

5.9948 

3i 

41 

8.511 2588 

246 

28 

5i 

32 

37 

76 

273 

29 

54 

33 

33 

64 

300 

30 

57 

34 

29 

52 

327 

3 1 

61 

35 

25 

40 

354 

32 

64 

36 

21 

28 

381 

33 

67 

37 

17 

l6 

408 

34 

70 

3 8 

13 

04 

434 

35 

73 

39 

09 

8.511 2492 

461 

36 

76 

40 

8.509 2505 

8.511 2480 

1.24488 

2.3637 

5.9980 

41 

OI 

68 

515 

38 

83 

42 

8.509 2497 

56 

542 

39 

86 

43 

93 

44 

569 

40 

89 

44 

89 

32 

595 

4» 

92 

45 

85 

20 

622 

42 

96 

46 

81 

08 

649 

43 

99 

47 

77 

8.511 2396 

676 

44 

6.0002 

48 

73 

84 

703 

44 

05 

49 

69 

72 

729 

45 

08 

50 

8.509 2465 

8.511 2360 

1.24756 

2.3646 

6.0011 

5i 

6t 

48 

783 

47 

*5 

52 

57 

35 

810 

48 

18 

53 

53 

23 

837 

49 

21 

54 

49 

II 

863 

50 

24 

55 

45 

8.511 2299 

890 

5* 

27 

56 

4 1 

87 

917 

52 

31 

57 

37 

75 

944 

53 

34 

58 

33 

63 

970 

54 

37 

59 

29 

5 1 

997 

55 

40 

60 

8.509 2425 

8.511 2239 j 

1.25024 

2.3656 

6.0043 


















654 COMPUTATION OF DISTANCES AND COORDINATES . 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 35°. 


Lat. 

log A 

diff. 1" = — 0.07 

log B 

diff. 1" = — 0.20 

log C 

diff. i"s= + °*44 

log D 

diff. 1 " = -f 0.01 

log 

diff. 1" = -f- 0.05 

O / 

35 oo 

8.509 24*5 

8.511 2239 

1.25024 

2.3656 

6.0043 

I 

21 

27 

050 

57 

47 

2 

17 

is 

077 

58 

5 ° 

3 

*3 

03 

104 

59 

53 

4 

09 

8.511 2191 

131 

59 

56 

05 

05 

78 

157 

60 

59 

6 

OI 

66 

184 

6l 

63 

7 

8.509 2396 

54 

211 

f>2 

66 

8 

93 

42 

237 

6 3 

69 

9 

88 

3 ° 

264 

64 

72 

IO 

8.509 2384 

8.511 2118 

1.25291 

2.3665 

6.0075 

11 

80 

06 

317 

66 

79 

12 

76 

8.511 2094 

344 

67 

82 

13 

72 

82 

371 

68 

85 

14 

68 

70 

397 

69 

88 

15 

64 

57 

424 

70 

9 i 

l6 

60 

45 

451 

70 

95 

*7 

56 

33 

477 

7 i 

98 

18 

52 

21 

504 

72 

6.oiox 

*9 

48 

09 

531 

73 

04 

20 

8.509 2344 

8.511 1997 

1-25557 

2 •3674 

6.0107 

21 

40 

85 

584 

75 

11 

22 

36 

72 

610 

76 

14 

23 

32 

60 

637 

77 

17 

24 

28 

48 

664 

78 

20 

25 

24 

36 

690 

79 

22 

26 

20 

24 

717 

79 

27 

27 

16 

12 

743 

80 

30 

28 

12 

OO 

770 

8l 

33 

29 

08 

8.511 1887 

796 

82 

36 

30 

8.509 2304 

8.511 1875 

1.25823 

2.3683 

6.0140 

31 

OO 

63 

850 

84 

43 

32 

8.509 2296 

51 

876 

85 

46 

33 

92 

39 

903 

86 

49 

34 

87 

27 

929 

86 

52 

35 

83 

15 

956 

87 

56 

36 

79 

02 

982 

88 

59 

37 

75 

8.511 1790 

1 .26009 

89 

62 

38 

7i 

78 

035 

90 

65 

39 

67 

66 

062 

9i 

69 

40 

8.509 2263 

8.511 1754 

x.26088 

2.3692 

6.0172 

41 

59 

4i 

115 

93 

75 

42 

55 

29 

141 

93 

78 

43 

5i 

17 

168 

94 

81 

44 

47 

05 

194 

95 

85 

45 

43 

8.511 1693 

221 

96 

88 

46 

39 

80 

247 

97 

9t 

47 

35 

68 

274 

98 

94 

48 

3i 

56 

300 

99 

98 

49 

27 

44 

327 

99 

6.0201 

50 

8.509 2222 

8.511 1632 

1.26353 

2.3700 

6.0204 

51 

18 

20 

380 

OT 

07 

52 

M 

07 

406 

02 

I I 

53 

IO 

8.5“ 1595 

432 

°3 

14 

54 

06 

83 

459 

04 

17 

55 

02 

71 

485 

05 

20 

56 

8.509 2x98 

58 

512 

05 

24 

57 

94 

46 

538 

06 

27 

58 

90 

34 

565 

07 

30 

59 

86 

22 

591 

08 

33 

60 

8.509 2182 

8.511 1510 

1.26617 

2.3709 

6.0237 



















FACTORS USED IN GEODETIC COMPUTATIONS. 655 


Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 

LATITUDE 36 °. 


Lat. 

log A 

log B 

diff. \" — — o.o7|diff. \" = — 0.20 

O / 



36 00 

3.509 2182 

8.511 1510 

I 

78 

8.511 1497 

2 

74 

85 

3 

70 

73 

4 

65 

6l 

05 

6l 

48 

6 

57 

36 

7 

53 

24 

8 

49 

12 

9 

45 

8.511 1399 

IO 

8.509 2141 

8.511 1387 

II 

37 

75 

12 

33 

63 

13 

29 

50 

*4 

25 

38 

15 

21 

26 

l6 

l6 

14 

17 

12 

OI 

18 

08 

8.511 1289 

19 

04 

77 

20 

8.509 2100 

8.511 1265 

21 

8.509 2096 

52 

22 

92 

40 

23 

88 

28 

24 

84 

15 

25 

80 

03 

26 

75 

8.511 1191 

27 

7 1 

79 

28 

67 

66 

29 

63 

54 

3 ® 

8.509 2059 

8.511 1142 

31 

55 

29 

32 

5 T 

17 

33 

47 

05 

34 

43 

8.511 1092 

35 

39 

80 

36 

35 

68 

37 

30 


38 

26 

43 

39 

22 

3 1 

4 ° 

41 

8.509 2018 

14 

8.5II IOI<) 

06 

42 

IO 

8.511 0994 

43 

44 

06 

02 

82 

69 

45 

8.509 1998 

57 

46 

93 

45 

47 

89 

32 

48 

49 

85 

81 

20 

08 

5 ° 

51 

8.509 1977 

73 

8.511 0895 

83 

52 

53 

54 

69 

65 

61 

71 

58 

46 

55 

56 

34 

56 

52 

21 

57 

48 

09 

58 

44 

8.511 0797 

84 

59 

40 

60 

8.509 1936 

8.511 0772 


log C 

log D 

log E 

1" = +0.44 

diff. 1" = -f- 0.01 

diff. i" = + o 

1.26617 

2.3709 

6.0237 

644 

10 

40 

670 

IO 

43 

697 

II 

46 

723 

12 

50 

749 

*3 

53 

776 

14 

56 

802 

*4 

59 

828 

15 

63 

855 

16 

66 

1.26881 

2.3717 

6.0269 

908 

18 

72 

934 

19 

76 

960 

19 

79 

987 

20 

82 

1.27013 

21 

85 

°39 

22 

89 

066 

23 

92 

092 

23 

95 

118 

24 

99 

1.27145 

2.3725 

6.0302 

171 

26 

05 

197 

27 

08 

223 

27 

12 

250 

28 

*5 

276 

29 

18 

302 

30 

21 

329 

3 1 

25 

355 

3 i 

28 

381 

32 

3 * 

1.27407 

2-3733 

6.0334 

434 

34 

38 

460 

35 

41 

486 

35 

44 

512 

36 

48 

539 

37 

5* 

565 

38 

54 

59 i 

38 

57 

617 

39 

6 l 

644 

40 

64 

1.27670 

2 - 374 * 

6.0367 

696 

4 1 

7* 

722 

42 

74 

00 10 

43 

44 

77 

80 

801 

45 

84 

827 

45 

87 

853 

46 

90 

879 

47 

94 

905 

48 

97 

1.27932 

2.3748 

6.0400 

958 

49 

03 

984 

50 

07 

1.28010 

5 * 

10 

036 

5 1 

*3 

062 

52 

*7 

088 

53 

20 

114 

54 

23 

141 

54 

27 

167 

55 

30 

1.28193 

2.3756 

6.0433 


























656 COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 37°. 


Lat. 

log A 

diff. \" = — 0.07 

log B 

diff. — — 0.21 

log C 

diff. i /, j= +0.43 

log D 

diff. 1" = +0.01 

log E 

diff. x" = -fo.06 

O / 

37 00 

8.509 1936 

8.511 0772 

1.28193 

2.3756 

6.0433 

I 

32 

60 

219 

56 

37 

2 

28 

47 

245 

57 

40 

'y 

23 

35 

271 

58 

43 

4 

19 

22 

297 

59 

46 

05 

15 

IO 

324 

59 

5 ° 

6 

II 

8.511 0698 

350 

60 

53 

7 

07 

85 

376 

6 l 

56 

8 

03 

73 

402 

62 

60 

9 

8.509 1899 

6 l 

428 

62 

63 

IO 

8.509 1895 

8.511 0648 

1.28454 

2.3763 

6.0466 

11 

90 

36 

480 

64 

70 

12 

86 

23 

506 

65 

73 

13 

82 

II 

532 

65 

76 

m 

78 

8.511 0599 

558 

66 

80 

15 

74 

86 

584 

67 

83 

l 6 

70 

74 

610 

67 

86 

17 

66 

6 l 

636 

68 

89 

18 

62 

49 

662 

69 

93 

19 

57 

37 

688 

69 

96 

20 

8.509 1853 

8.511 0524 

1.28715 

2.3770 

6.0499 

21 

49 

12 

74 i 

7 i 

6.0503 

22 

45 

OO 

767 

72 

06 

23 

4 i 

8.511 0487 

793 

72 

09 

24 

37 

75 

819 

73 

13 

25 

33 

62 

845 

74 

l 6 

26 

28 

50 

871 

74 

*9 

27 

24 

37 

897 

75 

23 

28 

20 

25 

923 

76 

26 

29 

l 6 

*3 

949 

76 

29 

30 

8.509 1812 

8.511 0400 

1.28975 

2-3777 

6-0533 

31 

08 

8.511 0388 

I .29001 

78 

36 

32 

04 

75 

02 7 

79 

39 

33 

OO 

63 

°53 

79 

43 

34 

8.509 1795 

5 i 

079 

80 

46 

35 

9 1 

38 

104 

81 

49 

36 

87 

26 

130 

81 

53 

37 

83 

13 

156 

82 

56 

38 

79 

OI 

l82 

83 

59 

39 

75 

8.511 0288 

208 

83 

63 

40 

8.509 1771 

8.511 0276 

1.29234 

2.3784 

6.0566 

4 i 

66 

64 

260 

85 

69 

42 

62 

5 1 

286 

85 

73 

43 

58 

39 

312 

86 

76 

44 

54 

26 

338 

87 

79 

45 

50 

M 

364 

87 

83 

46 

46 

OI 

39 ° 

88 

86 

47 

41 

8.511 0x89 

416 

89 

89 

48 

37 

76 

442 

89 

93 

49 

33 

64 

468 

90 

96 

50 

8.509 1729 

8.511 0151 

1.29494 

2 . 379 1 

6.0600 

5 1 

25 

39 

520 

9 i 

°3 

52 

21 

26 

546 

92 

06 

53 

16 

14 

57 i 

93 

TO 

54 

12 

02 

597 

93 

13 

55 

08 

8.5x1 0089 

623 

94 

16 

56 

04 

77 

649 

95 

20 

57 

00 

64 

675 

95 

23 

58 

8.509 1696 

52 

701 

96 

26 

59 

92 

39 

727 

96 

30 

60 

1_ 

8.509 1687 

8.511 0027 

*•29753 

2-3797 

6.0633 
























FACTORS USED IN GEODETIC COMPUTATIONS. 657 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 38°. 


































658 COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 39 °. 


Lat. 

log A 

diff. =— 0.07 

log B 

diff. 1" = — 0.21 

log C 

diff. 1"= +0.43 

log D 

diff. 1"= +0.01 

log E 

diff. r" — 0.06 

O / 

39 00 

8.509 1437 

8.510 9275 

1.31299 

2.3833 

6.0836 

I 

33 

62 

324 

33 

40 

2 

28 

50 

35 o 

34 

43 

3 

24 

37 

375 

35 

47 

4 

20 

25 

40 X 

35 

50 

05 

l6 

12 

427 

36 

53 

6 

12 

8.510 9199 

452 

3 6 

57 

7 

07 

87 

478 

37 

60 

8 

03 

74 

504 

37 

64 

9 

8.509 1399 

62 

529 

38 

67 

IO 

8.509 1395 

8.510 9149 

I- 3 I 555 

2.3838 

6.0871 

II 

91 

36 

581 

39 

74 

12 

86 

24 

606 

39 

77 

13 

82 

II 

632 

2.3840 

81 

14 

78 

8.510 9098 

658 

40 

84 

15 

74 

86 

683 

4 i 

88 

l6 

70 

73 

709 

4 i 

9 i 

17 

65 

61 

734 

42 

95 

18 

6l 

48 

760 

43 

98 

19 

57 

36 

786 

43 

6.0902 

20 

8.509 1353 

8.510 9023 

1.31811 

2.3844 

6.0905 

21 

49 

10 

837 

44 

08 

22 

44 

8.510 8998 

862 

45 

12 

*3 

40 

85 

888 

45 

15 

24 

36 

73 

9 i 3 

46 

19 

25 

32 

60 

939 

46 

22 

26 

28 

47 

965 

47 

26 

27 

23 

35 

990 

47 

29 

28 

*9 

22 

1.32016 

48 

32 

29 

15 

09 

041 

48 

36 

30 

8.509 1311 

8.510 8897 

1.32067 

2.3849 

6.0939 

31 

07 

84 

092 

49 

43 

32 

02 

72 

118 

2.3850 

46 

33 

8.509 1298 

59 

144 

50 

50 

34 

94 

46 

169 

51 

53 

35 

90 

34 

*95 

51 

57 

36 

86 

21 

220 

52 

60 

37 

81 

08 

246 

52 

63 

38 

77 

8.510 8796 

27I 

53 

67 

39 

73 

83 

297 

53 

70 

40 

8.509 1269 

8.510 8771 

* * 32323 

2.3854 

6.0974 

41 

64 

58 

348 

54 

77 

42 

60 

45 

374 

55 

81 

43 

5 6 

33 

399 

55 

84 

44 

52 

20 

425 

56 

88 

45 

48 

07 

450 

56 

9 1 

46 

43 

8.510 8695 

476 

57 

95 

47 

39 

82 

501 

57 

98 

48 

35 

69 

527 

57 

6.1002 

49 

3 1 

57 

552 

58 

05 

50 

8.509 1227 

8.510 8644 

1 -32578 

2.3858 

6.1008 

5 * 

22 

31 

603 

59 

12 

52 

18 

*9 

629 

59 

15 

53 

14 

06 

654 

2.3860 

19 

54 

IO 

8.510 8593 

680 

60 

22 

55 

06 

81 

705 

61 

26 

56 

01 

68 

731 

61 

29 

57 

8.509 1197 

55 

756 

62 

33 

r8 

93 

43 

782 

62 

36 

59 

89 

30 

807 

6 3 

40 

60 

8.509 1184 

8 510 8517 

1.32833 

2.3863 

6.1043 















FACTORS USED IN GEODETIC COMPUTATIONS. 659 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 40°. 





































66 O COMPUTATION OF DISTANCES AND COORD /NATES. 




Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 41 °. 


Lat. 

log 

A 

log 

B 

log C 

log D 

log E 

diff. 1" = 

- — 0.07 

diff. T' = 

= — 0.21 

diff. 1" = + 0.42 

diff. = +0.01 

diff. 1" = -j-0.06 

0 / 

41 OO 

8.509 

0930 

8.510 

7755 

I.34358 

2.3888 

6.1253 

I 


26 


42 

383 

88 

57 

2 


22 


30 

408 

89 

60 

3 


18 


17 

434 

89 

64 

4 


13 


04 

459 

89 

67 

05 


09 

8.510 

7691 

484 

90 

71 

6 


05 


79 

5io 

90 

75 

7 


OO 


66 

535 

90 

73 

8 

8.509 

0896 


53 

560 

91 

82 

9 

92 


40 

586 

9 i 

85 

TO 

8.509 

0888 

8.510 

7628 

1.34611 

2.3891 

6.1289 

1 1 

83 


15 

636 

92 

92 

12 


79 

8.510 

02 

662 

92 

96 

*3 


75 

7590 

687 

93 

99 

M 


7 * 


77 

712 

93 

6.1303 

15 


67 


64 

738 

93 

06 

l6 


62 


51 

763 

94 

IO 

17 


58 


39 

788 

94 

14 

18 


54 


26 

814 

94 

17 

19 


49 


13 

839 

95 

21 

20 

8.509 

0845 

8.510 

7500 

1.34864 

2.3895 

6.1324 

21 


41 

8.510 

7488 

890 

95 

28 

22 


37 


75 

9 l 5 

96 

31 

23 


32 


62 

940 

96 

35 

24 


28 


49 

965 

96 

38 

25 


24 


36 

991 

97 

42 

26 


20 


24 

1.35016 

97 

46 

27 


15 

8.510 

11 

041 

97 

49 

28 


I I 

7398 

066 

98 

53 

29 


07 


85 

092 

98 

56 

30 

8.509 

0803 

8.510 

7373 

I *35 I1 7 

2.3S98 

6.1360 

3* 

8.509 

0798 


60 

142 

99 

63 

32 


94 


47 

168 

99 

67 

33 


90 


34 

*93 

99 

70 

34 


86 


22 

218 

2.3900 

74 

35 


81 

8.510 

09 

243 

OO 

78 

36 


77 

7296 

269 

00 

81 

37 


73 


83 

294 

OO 

85 

38 


69 


70 

3i9 

OI 

88 

39 


64 


58 

345 

OI 

92 

40 

8.509 

0760 

8.510 

7245 

1-35370 

2.3901 

6.1395 

4i 


56 


32 

395 

02 

99 

42 

43 


52 

47 

8.510 

19 

07 

420 

446 

02 

02 

6.1403 

06 

44 


43 

7194 

47i 

03 

IO 

45 


39 


81 

496 

03 

13 

46 


35 


68 

522 

03 

17 

47 


30 


55 

547 

°3 

20 

48 


26 


43 

572 

04 

24 

49 


22 


30 

597 

04 

28 

50 

8.509 

0718 

8.510 

7117 

x.35623 

2.3904 

6.1431 

5i 


13 

8.510 

04 

648 

05 

35 

52 


09 

7091 

673 

05 

38 

53 


°5 


79 

698 

05 

42 

54 


OO 


66 

723 

05 

46 

55 

8.509 

0696 


53 

749 

06 

49 

56 


92 


40 

774 

06 

53 

57 


88 


27 

799 

06 

56 

58 


83 


15 

824 

07 

60 

59 


79 


02 

850 

07 

6 3 

60 

8.509 

0675 

8.510 

6989 

1 -35875 

2.3907 

6.1467 


















FACTORS USED IN GEODETIC COMPUTATIONS. 661 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 43°. 


Lat. 

log 

A 

log 

/>’ 

log C 

log D 

log E 

diff. 1 " = 

= “ 0.07 

diff. 1" = 

= — 0.21 

diff. x" = -j— 0.42 

diff. \" = -j-o.oo 

diff. x r/ = -f- 0.06 

O / 

42 OO 

8.509 

0675 

8.510 

6989 

1-35875 

2.3907 

6.1467 

I 


71 


76 

900 

08 

71 

2 


66 


64 

925 

08 

74 

3 


62 


51 

95 i 

08 

78 

4 


58 


38 

976 

08 

81 

05 


54 


25 

1.36001 

09 

85 

6 


49 


12 

026 

09 

89 

7 


45 


OO 

052 

09 

92 

8 


41 

8.510 

6887 

077 

09 

96 

9 


36 


74 

102 

10 

99 

10 

8.509 

0632 

8.510 

6861 

1.36127 

2.3910 

6.1503 

II 


28 


48 

152 

IO 

07 

12 


24 


36 

178 

IO 

IO 

13 


19 


23 

203 

II 

14 

14 


15 


IO 

228 

II 

*7 

is 


II 

8.510 

6797 

253 

II 

21 

l6 


07 


84 

278 

12 

25 

17 


02 


72 

304 

12 

28 

18 

OO 

Ln 

O 

vO 

0598 


59 

329 

12 

32 

19 


94 


46 

354 

12 

35 

20 

8.509 

6590 

8.510 

6733 

1.36379 

2 • 39 r 3 

6.1539 

21 


85 


20 

404 

13 

43 

22 


81 


07 

43 ° 

13 

46 

23 


77 

8 510 

6695 

455 

13 

50 

24 


72 


82 

480 

13 

54 

25 


68 


69 

5«5 

14 

57 

26 


64 


56 

530 

14 

6l 

27 


60 


43 

556 

14 

64 

28 


55 


31 

581 

14 

68 

29 


5 * 


18 

606 

15 

72 

3 ° 

8.509 

0547 

8.510 

6605 

1.36631 

2 . 39 x 5 

6.1575 

3 i 

43 

8.510 

6592 

656 

15 

79 

32 


38 


79 

682 

15 

83 

33 


34 


66 

707 

l6 

86 

34 


30 


54 

732 

16 

90 

35 


25 


41 

757 

l6 

93 

36 


21 


28 

782 

l6 

97 

37 


17 


15 

808 

17 

6.1601 

38 


13 


02 

833 

17 

04 

39 


08 

8.510 

6490 

858 

17 

08 

40 

8.509 

0504 

8.510 

6477 

1.36883 

2.3917 

6.1612 

41 

OO 


64 

908 

17 

15 

42 

8.509 

0496 


5 1 

934 

18 

T 9 

43 

9 l 


38 

959 

18 

22 

44 


87 


25 

984 

18 

26 

45 


83 


13 

1.37009 

18 

30 

46 


78 


OO 

034 

19 

33 

47 


74 

8.510 

6387 

059 

19 

37 

48 


70 


74 

085 

19 

4 i 

49 


66 


6l 

no 

i 9 

44 

50 

8.509 

0461 

8.510 

6348 

I.37135 

2.3919 

6.1648 

51 

57 


36 

160 

20 

52 

S 2 


53 


23 

185 

20 

55 

53 

54 


48 

44 

8.510 

IO 

6297 

210 

235 

20 

20 

59 

63 

55 

56 


40 

36 


84 

7i 

261 

286 

20 

21 

66 

70 

57 


3 1 


59 

311 

21 

73 

58 

59 


27 

23 


46 

33 

336 

361 

21 

21 

77 

81 

60 

8.509 

0419 

8.510 

6220 

I .37386 

2.3921 

6.1684 






















662 COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 43°. 


Lat. 

log /I 

diff. 1" = — 0.07 

log £ 

diff. 1 " — — 0.21 

log C 

diff. 1" = 0.42 

log D 

diff. 1" = +0.00 

log E 

diff. r" = -f-0.06 

O / 

43 °° 

8.509 0419 

8.510 6220 

1.37386 

2.3921 

6.1684 

i 

14 

07 

412 

22 

88 

2 

IO 

8.510 6195 

437 

22 

92 

3 

06 

82 

462 

22 

95 

4 

01 

69 

487 

22 

99 

05 

8.509 0397 

56 

512 

22 

6.1703 

6 

93 

43 

537 

22 

06 

7 

89 

30 

563 

23 

IO 

8 

84 

*7 

588 

23 

14 

9 

80 

05 

613 

23 

*7 

IO 

8.509 0376 

8.510 6092 

1.37638 

2.3923 

6.1721 

II 

7 i 

79 

663 

23 

25 

12 

67 

66 

688 

24 

28 

*3 

63 

53 

7 i 3 

24 

32 

14 

59 

40 

739 

24 

36 

15 

54 

28 

764 

24 

39 

16 

5 o 

15 

789 

24 

43 

17 

46 

02 

814 

24 

47 

>8 

4 i 

8.510 5989 

839 

25 

50 

*9 

37 

76 

864 

25 

54 

20 

8.509 0333 

8.510 5963 

1.37889 

2.3925 

6.1758 

21 

29 

50 

9*5 

25 

6l 

22 

24 

38 

940 

25 

6s 

23 

20 

25 

965 

25 

69 

24 

16 

12 

990 

25 

72 

25 

12 

8.510 5899 

1.38015 

26 

76 

26 

07 

86 

040 

26 

80 

27 

03 

73 

065 

26 

83 

28 

8.509 0299 

60 

O9I 

26 

87 

29 

94 

48 

116 

26 

9 i 

30 

8.509 0290 

8.510 5835 

1.38141 

2.3926 

6.1795 

31 

86 

22 

166 

27 

98 

32 

82 

09 

. 191 

27 

6.1802 

33 

77 

8.510 5796 

216 

27 

06 

34 

73 

83 

241 

27 

09 

35 

69 

71 

266 

27 

13 

36 

64 

58 

292 

27 

17 

37 

60 

45 

317 

27 

20 

38 

56 

32 

342 

27 

24 

39 

52 

19 

367 

28 

28 

40 

8.509 0247 

8.510 5706 

1.38392 

2.3928 

6.1831 

4 * 

43 

8 . 5 ig 5693 

4 i 7 

28 

35 

42 

39 

81 

442 

28 

39 

43 

34 

68 

467 

28 

42 

44 

30 

55 

492 

28 

46 

45 

26 

42 

518 

28 

50 

46 

22 

29 

543 

28 

53 

47 

17 

l6 

568 

29 

57 

48 

13 

03 

593 

29 

6l 

49 

09 

8.510 5591 

618 

29 

65 

50 

8.509 0204 

8.510 5578 

1.38643 

2.3929 

6.1868 

5 1 

OO 

65 

668 

29 

72 

52 

8.509 0196 

52 

693 

29 

76 

53 

92 

39 

719 

29 

79 

54 

87 

26 

744 

29 

83 

55 

83 

*3 

769 

30 

87 

56 

79 

OI 

794 

30 

9 i 

57 

74 

8.510 5488 

819 

30 

94 

58 

70 

75 

844 

30 

98 

59 

66 

62 

869 

30 

6.1902 

60 

8.509 0162 

8.510 5449 

1.38894 

2.3930 

6.1905 

- - 















FACTORS USED IN GEODETIC COMPUTATIONS. 663 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 44°. 


Lat. 

log A 

log B 

log C 

log D 

log E 

diff. — — 0.07 

diff. \" — — o.2i 

diff. 1" = -4-0.42 

lilt. 1" = -f* °.oo 

diff. 1" = -f-0.06 

O / 

44 00 

8.509 0162 

8.510 5449 

1.38894 

2.3930 

6.1905 

1 

57 

36 

919 

30 

09 

2 

53 

23 

945 

30 

J 3 

3 

49 

01 

970 

30 

17 

4 • 

44 

8.510 5388 

995 

30 

20 

05 

40 

75 

1.39020 

3 1 

24 

6 

36 

62 

045 

3 i 

28 

7 

31 

49 

070 

3 1 

31 

8 

27 

36 

095 

3 1 

35 

9 

23 

23 

120 

3 i 

39 

IO 

8.509 0119 

8.510 5311 

I- 39 M 5 

2.3931 

6.1943 

II 

14 

07 

171 

3 1 

46 

12 

IO 

8.510 5295 

196 

3 1 

5 ° 

13 

06 

82 

221 

3 * 

54 

M 

02 

69 

246 

3 * 

58 

15 

8.509 0097 

56 

27I 

31 

6l 

l6 

93 

43 

296 

3 i 

65 

17 

89 

30 

321 

32 

69 

18 

84 

18 

346 

32 

72 

19 

80 

05 

371 

32 

76 

20 

8.509 0076 

8.510 5192 

1 • 39396 

2.3932 

6.1980 

21 

72 

79 

422 

32 

84 

22 

67 

66 

447 

32 

87 

23 

63 

53 

472 

32 

9 i 

24 

59 

40 

497 

32 

95 

25 

54 

28 

522 

32 

99 

26 

5 ° 

1 5 

547 

32 

6.2002 

27 

46 

02 

572 

32 

06 

28 

42 

8.510 5089 

597 

32 

IO 

29 

37 

76 

623 

32 

*4 

3 ° 

8.509 0033 

8.510 5063 

1.39648 

2.3932 

6.2017 


29 

50 

673 

32 

21 

32 

24 

37 

698 

32 

25 

OO 

20 

25 

723 

33 

29 

34 

16 

12 

748 

33 

32 


II 

8.510 4999 

773 

33 

36 

36 

07 

86 

798 

33 

40 

37 

38 

03 

8.508 9999 

73 

60 

823 

848 

33 

33 

44 

47 

39 

94 

47 

873 

33 

5 1 

4 ° 

8.508 9990 

8.510 4935 

1.39898 

2-3933 

6.2055 

41 

86 

22 

924 

33 

59 

42 

81 

09 

949 

33 

62 

43 

77 

8.510 4896 

974 

33 

66 

44 

73 

83 

999 

33 

70 

45 

69 

70 

1.40024 

33 

74 

46 

64 

57 

049 

33 

77 


60 

44 

074 

33 

oi 

4 / 

48 

49 

56 

51 

32 

19 

099 

124 

33 

33 

85 

89 

5 ° 

Cl 

8.508 9947 

43 

8.510 4806 
8.510 4793 

1.40149 

174 

2-3933 

33 

6.2092 

96 

52 

53 

54 

39 

34 

30 

80 

67 

54 

200 

225 

250 

33 

33 

33 

6.2100 

04 

08 

CC 

26 

41 

275 

33 

II 

s 6 

21 

29 

3 °° 

33 

»5 

57 

s8 

17 

*3 

16 

03 

325 

35 ° 

33 

33 

19 

23 

59 

09 

8.510 4690 

375 

33 

27 

60 

l-- 

8.508 9904 

8.510 4677 

1.40400 

2-3933 

1 6.2130 





































664 COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVII. 


FACTORS USED IN GEODETIC COMPUTATIONS. 

LATITUDE 45° 


Lat. 

log A 

diff. 1" = — 0.07 

log B 

diff. \" — - 0.21 

log C 

diff. 1"= +0.42 

log D 

diff. — ± 0.00 

log E 

diff. — -f- 0.06 

O / 

45 00 

8.508 9904 

8.510 4677 

1.40400 

2-3933 

6.2130 

1 

OO 

64 

425 

33 

34 

2 

8.508 9896 

51 

450 

33 

38 

3 

91 

39 

475 

34 

42 

4 

87 

26 

501 

34 

46 

05 

83 

13 

526 

34 

49 

6 

78 

OO 

55 i 

34 

53 

7 

74 

8.510 4587 

576 

34 

57 

8 

70 

74 

601 

34 

6l 

9 

66 

6l 

626 

34 

64 

IO 

8.508 9861 

8.510 4548 

1.40651 

2-3934 

6.2168 

IT 

57 

36 

676 

34 

72 

12 

53 

23 

701 

34 

76 

13 

48 

IO 

727 

34 

80 

14 

44 

8.510 4497 

752 

34 

83 

15 

40 

84 

777 

33 

87 

16 

3 6 

7 i 

802 

33 

9 i 

17 

3 1 

59 

827 

33 

95 

18 

27 

46 

852 

33 

99 

19 

23 

33 

877 

33 

02 

20 

8.508 9818 

8.510 4420 

1.40902 

2 3933 

6.2206 

21 

M 

07 

927 

33 

IO 

22 

IO 

8.510 4394 

952 

33 

14 

23 

06 

81 

978 

33 

18 

24 

OI 

68 

1.41003 

33 

21 

25 

8.508 9797 

56 

028 

33 

25 

26 

93 

43 

053 

33 

29 

27 

88 

30 

078 

33 

33 

28 

84 

17 

103 

33 

37 

29 

80 

04 

128 

33 

40 

30 

8.508 9776 

8.510 4291 

1 . 4”53 

2-3933 

6.2244 

31 

7 i 

78 

178 

33 

48 

32 

67 

65 

203 

33 

52 

33 

63 

52 

229 

33 

56 

34 

58 

40 

254 

33 

60 

35 

54 

27 

279 

33 

63 

36 

50 

M 

304 

33 

67 

37 

46 

OI 

329 

33 

71 

38 

4 i 

8.510 4188 

354 

33 

75 

39 

37 

75 

379 

33 

79 

40 

8.508 9733 

8.510 4162 

1.41404 

2-3933 

6.2283 

4 i 

28 

49 

429 

33 

86 

42 

24 

37 

454 

33 

90 

43 

20 

24 

479 

33 

94 

44 

l6 

11 

505 

33 

98 

45 

IT 

8.510 4098 

53 ° 

33 

6.2302 

46 

07 

85 

555 

32 

06 

47 

03 

72 

580 

32 

09 

48 

8.508 9698 

60 

605 

32 

13 

49 

94 

47 

630 

32 

17 

50 

8.508 9690 

8.510 4034 

1-41655 

2.3932 

6.2321 

5 i 

86 

21 

680 

32 

25 

52 

82 

08 

705 

32 

29 

53 

78 

8.510 3995 

73 i 

32 

32 

54 

74 

82 

756 

32 

36 

55 

68 

69 

781 

32 

40 

56 

64 

57 

806 

32 

44 

57 

60 

44 

831 

32 

48 

58 

55 

31 

856 

32 

52 

59 

5 1 

18 

881 

32 

55 

60 

8.508 9647 

8.510 3905 

1.41906 

2 - 3932 

6.2359 















FACTORS USED IN GEODETIC COMPUTATIONS. 665 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 4 «° 


/ 

T at 

log 

A 

log B 

log C 

log D 

log E 


diff. 1" = 

= — 0.07 

diff. x" = — 0.21 

diff. x"— + 0.42 

diff. \" — — 0.00 

diff. \" — -f- 0.06 

O / 

46 00 

8.508 

9647 

8.510 3905 

1.41906 

2• 3932 

6.2359 

I 


43 

8.510 3892 

93 i 

32 

63 

2 


38 

79 

957 

3 i 

67 

3 


34 

67 

982 

31 

71 

4 


3 ° 

54 

1.42007 

3 i 

75 

05 


25 

41 

032 

31 

79 

6 


21 

28 

057 

3 i 

82 

7 


17 

15 

082 

31 

86 

8 


13 

02 

107 

31 

90 

9 


08 

8.51° 3789 

132 

31 

94 

IO 

8.508 

9604 

8.510 3776 

1-42157 

2 - 393 * 

6.2398 

11 

OO 

64 

183 

3 i 

6.2402 

12 

8.508 

9595 

5 i 

208 

3 i 

06 

13 


91 

38 

233 

30 

09 

14 


87 

25 

258 

30 

13 

15 


83 

12 

283 

30 

17 

l6 


78 

8.510 3699 

308 

3 ° 

21 

u 


74 

86 

333 

30 

25 

18 


70 

74 

358 

3 ° 

29 

19 


65 

61 

384 

30 

33 

20 

8.508 

95 6 i 

8.510 3648 

I .42409 

2• 3930 

6.2436 

21 


57 

35 

434 

30 

40 

22 


53 

22 

459 

30 

44 

23 


48 

09 

484 

29 • 

48 

24 


44 

8.510 3596 

509 

29 

52 

25 


40 

84 

534 

29 

56 

26 


35 

71 

559 

29 

60 

27 


3 1 

58 

584 

29 

64 

28 


27 

45 

610 

29 

67 

29 


23 

32 

635 

29 

71 

3 ° 

8.509 

95 l8 

8.510 3519 

1.42660 

2.3929 

6.2475 

3 l 

14 

06 

685 

29 

79 

32 


IO 

8.510 3494 

710 

28 

83 

33 


05 

81 

735 

28 

87 

34 


OI 

68 

700 

28 

91 

35 

8.508 

9497 

55 

. 786 

28 

95 

36 

93 

42 

811 

28 

99 



88 

29 

836 

28 

6.2502 

38 


84 

17 

861 

28 

06 

39 


80 

04 

886 

28 

10 

40 

8.508 

9475 

8.510 3391 

1.42911 

2.3927 

6.2514 

41 

7 1 

78 

936 

27 

iS 

42 


67 

65 

961 

27 

22 

43 


63 

52 

987 

27 

26 

44 


58 

39 

1.43012 

27 

3 ° 

45 


54 

27 

037 

27 

34 

46 


5 ° 

r 4 

062 

27 

38 

47 


45 

OI 

087 

26 

4 i 

48 


41 

8.510 3288 

112 

26 

45 

49 


37 

75 

137 

26 

49 

50 

51 

52 

53 

54 

8.508 

9433 

28 

24 

20 

16 

8.510 3262 

49 

37 

24 

11 

1.43163 

188 

213 

238 

263 

2.3926 

26 

26 

26 

25 

6.2553 

57 

61 

65 

69 

55 


11 

8.510 3198 

288 

25 

73 

56 


07 

85 

314 

25 

77 

57 

58 

59 

8.508 

03 

9398 

94 

72 

60 

47 

339 

3 6 4 

389 

35 

25 

25 

84 

88 

60 

8.508 

9390 

8.510 3134 

I- 434 I 4 

2.3924 

6.2592 

-- — U 
























666 COMPUTATION OF DISTANCES AND COORDINATES. 


Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 

LATITUDE 47°. 


Lat. 

log A 

diff. \" — — 0.07 

log Z? 

diff. 1" = — 0.21 

log C 

diff. 1" = -{-0.42 

log D 

diff. = -j- 0.00 

log E 

diff. = -|- o.Oj 

o / 

47 00 

8.508 9390 

8.510 3134 

1.43414 

2.3924 

6.2592 

I 

86 

21 

439 

24 

96 

2 

81 

08 

465 

24 

6.2600 

3 

77 

8.510 3095. 

490 

24 

04 

4 

73 

82 

5*5 

24 

08 

05 

68 

70 

540 

24 

12 

6 

64 

57 

565 

23 

16 

7 

60 

44 

590 

23 

20 

8 

56 

31 

615 

23 

24 

9 

5i 

18 

641 

23 

28 

io 

8.508 9347 

8.510 3005 

1.43666 

2.3923 

6.2632 

11 

43 

8.510 2993 

691 

23 

35 

12 

38 

80 

716 

22 

39 

13 

34 

67 

74i 

22 

43 

14 

30 

54 

766 

22 

47 

15 

26 

41 

792 

22 

51 

l6 

21 

28 

817 

22 

55 

17 

17 

l6 

842 

21 

59 

18 

*3 

°3 

867 

21 

63 

19 

09 

8.510 2890 

892 

21 

67 

20 

8.508 9304 

8.510 2877 

I-439I7 

2.3921 

6.2671 

21 

OO 

64 

943 

21 

75 

22 

8.508 9296 

51 

968 

20 

79 

23 

91 

39 

993 

20 

83 

24 

87 

26 

1.44018 

20 

87 

25 

83 

*3 

043 

20 

9i 

26 

79 

OO 

069 

20 

95 

27 

74 

8.510 2787 

094 

19 

99 

28 

70 

74 

119 

19 

6.2702 

29 

66 

62 

144 

>9 

06 

30 

8.508 9261 

8.510 2749 

1.44169 

2 -39 J 9 

6.2710 

31 

57 

36 

195 

19 

14 

32 

53 

23 

220 

18 

18 

33 

49 

IO 

245 

18 

22 

34 

44 

8.510 2698 

270 

18 

26 

35 

40 

85 

295 

18 

30 

36 

36 

72 

321 

18 

34 

37 

32 

59 

346 

17 

38 

38 

27 

46 

371 

17 

42 

39 

23 

33 

396 

17 

46 

40 

8.508 9219 

8.510 2621 

1.44421 

2 .39U 

6.2750 

41 

*4 

08 

447 

l6 

54 

42 

IO 

8.510 2595 

472 

l6 

58 

43 

06 

82 

497 

l6 

62 

44 

02 

69 

522 

l6 

66 

45 

8.508 9197 

57 

547 

l6 

70 

46 

93 

44 

573 

15 

74 

47 

89 

3 1 

598 

15 

78 

48 

84 

18 

623 

15 

82 

49 

80 

05 

648 

>5 

86 

50 

8.508 9176 

8.510 2493 

1-44673 

2 -39i4 

6.2790 

51 

72 

80 

699 

M 

94 

52 

67 

67 

724 

14 

98 

53 

63 

43 

749 

M 

6.2802 

54 

59 

4i 

774 

*3 

06 

55 

55 

28 

800 

13 

10 

56 

50 

l6 

825 

13 

14 

57 

49 

03 

850 

1.3 

18 

58 

42 

8.510 2390 

875 

12 

22 

59 

38 

77 

900 

12 

26 

60 

8.508 9133 

8.510 2364 

1.44926 

j 2.3912 

6.2830 






























FACTORS USED IN GEODETIC COMPUTATIONS . 667 


Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 48°. 


Lat. 

log 

diff. 1" = - 0.07 

log B 

diff. 1" = — 0.21 

log C 

diff. — -f-0.42 

log D 

diff. \" — — 0.00 

log £ 

diff. 1" = +0.07 

O / 

48 00 

8.508 9133 

8.510 2364 

1.44926 

2.3912 

6.2830 

I 

29 

52 

95 i 

12 

34 

2 

25 

39 

976 

II 

38 

3 

20 

20 

1.45001 

II 

42 

4 

l6 

13 

027 

II 

46 

05 

12 

OO 

052 

11 

50 

6 

08 

8.510 2288 

077 - 

IO 

54 

7 

03 

75 

102 

IO 

58 

8 

8.508 9099 

62 

128 

IO 

62 

9 

95 

49 

153 

10 

66 

IO 

8.508 9091 

8.510 2236 

1.45178 

2.3909 

6.2870 

II 

86 

24 

203 

09 

74 

12 

82 

II 

229 

09 

78 

13 

78 

8.510 2198 

254 

08 

82 

14 

74 

85 

279 

08 

86 

*5 

69 

72 

304 

08 

90 

l6 

65 

60 

330 

08 

94 

17 

61 

47 

355 

07 

98 

18 

57 

34 

380 

07 

6.2902 

19 

52 

21 

406 

07 

06 

20 

8.508 9048 

8.510 2108 

1-45431 

2.3907 

6.2910 

21 

44 

8.510 2096 

456 

06 

M 

22 

39 

83 

481 

06 

18 

23 

35 

70 

507 

06 

22 

24 

3 i 

57 

532 

05 

26 

25 

27 

45 

557 

05 

30 

26 

22 

32 

582 

05 

34 

27 

18 

19 

608 

os 

38 

28 

14 

06 

633 

04 

42 

29 

10 

8.510 1993 

658 

04 

46 

30 

8.508 9005 

8.510 1981 

1.45683 

2.3904 

6.2950 

3 1 

or 

68 

709 

03 

54 

32 

8.508 8997 

55 

734 

03 

58 

33 

93 

42 

759 

03 

62 

34 

88 

30 

785 

02 

66 

35 

84 

17 

810 

02 

70 

36 

80 

04 

835 

02 

74 

37 

76 

8.510 1891 

861 

02 

78 

38 

7 * 

78 

886 

OI 

82 

39 

67 

66 

9 ” 

01 

86 

40 

8.508 8963 

8.510 1853 

1 -45937 

2.3901 

6.2990 

41 

59 

40 

962 

OO 

94 

42 

54 

27 

987 

OO 

98 

43 

50 

15 

1.46012 

OO 

6.3002 

44 

46 

02 

038 

2.3899 

06 

45 

4i 

8.510 1789 

063 

99 

IO 

46 

37 

76 

088 

99 

15 

47 

33 

. 64 

114 

98 

*9 

48 

29 

51 

139 

98 

23 

49 

24 

38 

164 

98 

27 

5 ° 

8.508 8920 

8.510 1725 

1.46190 

2.3897 

6.3031 

51 

l6 

13 

215 

97 

35 

52 

12 

OO 

240 

97 

39 

53 

08 

8.510 1687 

266 

96 

43 

54 

03 

74 

291 

96 

47 

55 

8.508 8899 

62 

316 

96 

5 1 

56 

95 

49 

342 

95 

55 

57 

90 

36 

3 6 7 

95 

59 

58 

86 

*3 

392 

95 

63 

59 

82 

IO 

418 

94 

67 

60 

8.508 8878 

8.510 1598 

1.46443 

2.3894 

6.3071 

















668 COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVI r. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 49". 


Lat. 

log A 

j diff. x " = — 0.07 

lo £ B 

diff. 1" = — 0.21 

log c 

diff. x " = -j -0 .4 2 

log D 

diff. x " = — 0.01 

log E 

diff. x " = -f- 0.07 

o / 

49 oo 

8.508 8878 

8.510 1598 

1.46443 

2.3894 

6.3071 

i 

73 

85 

468 

94 

75 

2 

69 

72 

494 

93 

79 

3 

65 

59 

5i9 

93 

84 

4 

6l 

47 

544 

93 

86 

05 

57 

34 

57° 

92 

92 

6 

52 

21 

595 

92 

96 

7 

48 

08 

621 

92 

6.3100 

8 

44 

8.510 1496 

646 

9 1 

04 

9 

39 

83 

671 

9i 

08 

IO 

8.508 8835 

8.510 1470 

1.46696 

2.3891 

6.3112 

II 

3i 

58 

722 

90 

l6 

12 

27 

45 

747 

90 

20 

13 

23 

32 

773 

2.3889 

24 

14 

18 

T 9 

798 

89 

28 

15 

M 

07 

824 

89 

32 

l6 

IO 

8.510 1394 

849 

88 

37 

17 

06 

81 

874 

88 

41 

18 

01 

68 

899 

88 

45 

19 

8.508 8797 

56 

925 

87 

49 

20 

8.508 8793 

8.510 1343 

1.46950 

2.3887 

6.3153 

21 

89 

30 

976 

87 

57 

22 

84 

17 

I.47OOI 

86 

6l 

23 

80 

05 

026 

86 

65 

24 

76 

8.510 1292 

052 

85 

69 

25 

72 

79 

077 

85 

73 

26 

67 

67 

103 

85 

78 

27 

63 

54 

128 

84 

82 

28 

59 

4 i 

153 

84 

86 

29 

55 

28 

179 

83 

90 

30 

8.508 8750 

8.510 1216 

1.47204 

2.3883 

6.3194 

31 

46 

03 

230 

83 

98 

32 

42 

8.510 1190 

255 

82 

6.3202 

33 

38 

78 

281 

82 

06 

34 

33 

65 

306 

82 

IO 

35 

29 

52 

33 i 

81 

15 

36 

25 

39 

357 

81 

19 

37 

21 

27 

382 

80 

23 

38 

l6 

14 

408 

80 

27 

39 

12 

OI 

433 

80 

31 

40 

8.508 8708 

8.510 1088 

1 -47459 

2.3879 

6.3235 

41 

04 

77 

484 

79 

39 

42 

OO 

63 

5°9 

78 

43 

43 

8.508 8695 

50 

535 

78 

47 

44 

9 1 

38 

560 

78 

52 

45 

87 

25 

586 

77 

56 

46 

83 

12 

611® 

77 

60 

47 

78 

OO 

637 

76 

64 

48 

74 

8.510 0987 

662 

76 

68 

49 

70 

74 

688 

75 

72 

50 

8.508 8666 

8.510 0962 

1 - 477 I 3 

2-3875 

6.3276 

51 

61 

49 

738 

75 

81 

52 

57 

36 

764 

74 

85 

53 

53 

23 

789 

74 

89 

54 

49 

II 

815 

73 

93 

55 

45 

8.510 0898 

840 

73 

97 

56 

40 

85 

866 

73 

6.3301 

57 

36 

73 

891 

72 

05 

58 

32 

60 

917 

72 

09 

59 

28 

48 

942 

7i 

14 

60 

8.508 8623 

8.510 0835 

1.47968 

8.3871 

6.3318 















FACTORS USED IN GEODETIC COMPUTATIONS. 669 

Table XXXVII. 

FACTORS USED IN GEODETIC COMPUTATIONS. 


LATITUDE 50°. 


Lat. 

log /i 

diff. 1" = — 0.07 

log B 

liff. 1" — — 0.21 c 

log C 

liff. \" = — 0.43 

log D 

liff. 1 ' — — 0.01 c 

log E 

liff. 1" = -I-0.07 


• / 

50 00 

1 

2 

3 

4 

OS 

6 

7 

8 

9 

10 

11 

12 

13 

14 

*5 

16 

17 

18 

*9 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

8.508 8623 

19 

15 

11 

06 

02 

8.508 8598 

94 

90 

85 

8.508 8581 

77 

75 

68 

64 

60 

56 

52 

47 

43 

8.508 8539 

35 

30 

26 

22 

18 

14 

09 

05 

01 

8.508 8497 

93 

88 

84 

80 

76 

7 1 

67 

63 

59 

8.508 8455 

50 

46 

42 

38 

34 

29 

25 

21 

*7 

8.508 8413 

08 

04 

00 

8.508 8396 

92 

87 

83 

79 

75 

8.508 8371 
_1----- 

8.510 0835 

22 

09 

8.510 0797 

84 

71 

59 

46 

33 

21 

8.510 0708 
8.510 0695 

83 

70 

57 

45 

32 

i9 

07 

8.510 0594 

8.510 0581 

69 

56 

43 

3* 

18 

05 

8.510 0493 

80 

67 

8.510 0455 

42 

29 

*7 

04 

8.510 0392 

79 

66 

54 

4 1 

8.510 0328 

16 

03 

8.510 0291 

78 

65 

53 

40 . 

27 

15 

8.510 0202 
8.510 0190 

77 

64 

52 

39 

27 

14 

01 

8.510 0089 
8.510 0076 

1.47968 

993 

1.48019 

044 

070 

095 

121 

146 

172 

197 

1.48223 

248 

274 

299 

325 

35° 

376 

401 

427 

452 

1.48478 

5°4 

529 

555 

580 

606 

631 

657 

682 

708 

1.48734 

759 

785 

810 

836 

861 

887 

9i3 

938 

964 

1.48989 

1•49 oi 5 

041 

066 

092 

117 

M3 

169 

194 

220 

1.49246 

271 

297 

322 

348 

374 

399 

425 

45i 

476 

1.49502 

2.3871 

70 

70 

70 

69 

69 

68 

68 

67 

67 

2.3866 

66 

66 

65 

65 

64 

64 

63 

63 

62 

2.3862 

61 

61 

60 

60 

60 

59 

59 

58 

58 

2.3857 

57 

56 

56 

55 

55 

54 

54 

53 

53 

2.3852 

52 

51 

5i 

5° 

50 

49 

49 

48 

48 

2.3847 

47 

46 

46 

45 

45 

44 

44 

43 

43 

2.3842 

6.3318 

22 

26 

30 

34 

39 

43 

47 

5 T 

55 

6-3359 

63 

68 

72 

76 

80 

84 

88 

93 

97 

6.3401 

05 

09 

14 

18 

22 

26 

30 

34 

39 

6.3443 

47 

5i 

55 

60 

64 

68 

72 

76 

81 

6.3485 

89 

93 

97 

6.3502 

06 

10 

14 

18 

23 

6.3527 

31 

35 

40 

44 

48 

52 

56 

61 

65 

6.3569 



























6/0 COMPUTATION OF DISTANCES AND COORDINATES. 

Table XXXVIII. 

CORRECTIONS TO LONGITUDE FOR DIFFERENCE IN ARC 

AND SINE. 

(From Appendix No. 9, Report of U. S. Coast and Geodetic Survey, 1894.) 


Lo^ N(-) 

Log Difference. 

Log A\ (+) 

Log A (-) 

Log Difference. 

Log A A. (+) 

3 876 

0.000 0001 

2.385 

5-010 

0.000 0186 

3 - 5 x 9 

4.026 

02 

2-535 

5- OI 7 

192 

3.526 

4 i >4 

03 

2.623 

5-025 

199 

3-534 

4.177 

04 

2.686 

5-033 

206 

3 • 542 

4.225 

05 

2-734 

5.040 

213 

3-549 

4.265 

c6 

2.774 

5-047 

221 

3.556 

4.298 

07 

2.807 

5-054 

228 

3-563 

4-327 

08 

2.836 

5.062 

236 

3 - 57 1 

4-353 

09 

2.862 

3.068 

243 

3 577 

4.376 

10 

2.885 

5-075 

25 x 

3-584 

4-396 

11 

2.905 

5.082 

259 

3 - 59 ' 

4-415 

12 

2.924 

5.088 

267 

3-597 

4-433 

13 

2.942 

5-095 

275 

3.604 

4-449 

14 

2.958 

5.102 

284 

3.611 

4.464 

15 

2.973 

5.108 

292 

3 - 6 i 7 

4.478 

. 16 

2.987 

5 - XI 4 

300 

3.623 

4.491 

17 

3.000 

5.120 

309 

3.629 

4-503 

18 

3.012 

5.126 

318 

3-635 

4 526 

20 

3-035 

5 X 32 

327 

3.641 

4 548 

23 

3-057 

5 - x 38 

336 

3-647 

4-570 

25 

3-079 

5 - x 44 

345 

3-653 

4-591 

27 

3.100 

5 - x 5 ° 

354 

3-659 

4.612 

30 

3 - x 2 I 

5- x 56 

364 

3-665 

4.631 

33 

3.140 

5 - x 6 i 

373 

3.670 

4.649 

36 

3 - x s8 

5- x 67 

383 

3.676 

4.667 

39 

3- x 76 

5- x 72 

392 

3-68i 

4.684 

42 

3- x 93 

5- x 78 

402 

3.687 

4.701 

45 

3.210 

5-183 

412 

3.692 

4.716 

48 

3-225 

5.188 

422 

3-697 

4-732 

52 

3 - 2 4 x 

5 - x 93 

433 

3.702 

4.746 

56 

3-255 

5- x 99 

443 

3.708 

4.761 

59 

3.270 

5-204 

453 

3.713 

4-774 

63 

3.283 

5.209 

464 

3 - 7 i 8 

4.788 

67 

3-297 

5-214 

474 

3-723 

4.801 

7 l 

3.310 

5.219 

486 

3-728 

4.813 

75 

3-322 

5.223 

497 

3 - 73 2 

4.825 

80 

3-334 

5.228 

508 

3-737 

4-834 

84 

3-343 

5-233 

5 X 9 

3.742 

4.849 

89 

3-358 

5-238 

530 

3-747 

4.860 

94 

3-369 

5.242 

54 x 

3 - 75 1 

4.871 

98 

3 - 38 o 

5-247 

553 

3-756 

4.882 

103 

3 - 39 x 

5-251 

565 

3.760 

4.892 

108 

3.401 

5-256 

577 

3-765 

4 - 9°3 

114 

3-412 

5.260 

588 

3.769 

4 - 9 I 3 

xx 9 

3-422 

5-265 

600 

3-774 

4.922 

124 

3 - 43 1 

5.269 

6 X 3 

3-778 

4.932 

x 3 o 

3-441 

5-273 

625 

3.782 

4.941 

136 

3-450 

5.278 

637 

3.787 

4-950 

142 

3-459 

5.282 

650 

3.791 

4-959 

x 47 

3.468 

5.286 

663 

3-795 

4.968 

x 53 

3-477 




4.976 

160 

3.485 




4-985 

166 

3-494 




4-993 

172 

3-502 




5.002 

x 79 

3 - 5 11 









































FACTORS USED IN GEODETIC COMPUTATIONS. 671 


Table XXXIX. 

VALUES OF LOG ---. 

cos \ d(p 


(From Appendix No. 9, Report of U. S. Coast and Geodetic Survey, 1894.) 


--—— 

> 

log. sec. 1— 1. 

Ac/>. 

log. sec.| A ^| 

A < f >. 

log. sec. 1— 1. 

io' 

0.000 000 

40' 

0.000 007 

70 

0.000 022 

II 

I 

4i 

8 

71 

23 

12 

I 

42 

8 

72 

24 

13 

I 

43 

8 

73 

24 

14 

I 

44 

9 

74 

25 

15 

I 

45 

9 

75 

26 

l6 

I 

46 

IO 

76 

26 

17 

I 

47 

IO 

77 

27 

18 

I 

48 

II 

78 

28 

19 

2 

49 

II 

79 

29 

20 

2 

50 

II 

80 

29 

21 

2 

5i 

12 

81 

30 

22 

2 

52 

12 

82 

31 

23 

2 

53 

13 

83 

32 

24 

3 

54 

13 

84 

32 

25 

3 

55 

14 

85 

33 

26 

3 

56 

*4 

86 

34 

27 

3 

57 

15 

87 

35 

28 

4 

58 

15 

88 

36 

29 

4 

59 

l6 

89 

36 

30 

4 

60 

16 

90 

37 

31 

4 

6l 

17 

9 i 

38 

32 

5 

62 

18 

92 

39 

33 

5 

63 

18 

93 

40 

34 

5 

64 

19 

94 

41 

35 

6 

65 

*9 

95 

41 

36 

6 

66 

20 

96 

42 

37 

6 

67 

21 

97 

43 

38 

7 

68 

21 

98 

44 

39 

7 

69 

22 

99 

45 


Table XL. 

LOG F. 

(From Appendix No. 9, Report of U. S. Coast and Geodetic Survey, 1894.) 


Lat. 

Log A'. 

Lat. 

Log A. 

Lat. 

Log. A. 

Lat. 

Log A. 

2 3° 

C* 

M 

00 

£ 

34° 

7.877 

45° 

7.840 

56° 

7.706 

24 

23 

35 

77 

46 

32 

57 

7.688 

25 

32 

36 

77 

47 

24 

58 

69 

26 

41 

37 

76 

48 

14 

59 

49 

27 

49 

38 

74 

49 

04 

60 

27 

28 

55 

39 

72 

50 

7.792 

61 

05 

29 

61 

40 

69 

5i 

80 

62 

7-58 i 

3° 

66 

4 1 

64 

52 

67 

63 

56 

3 1 

70 

42 

60 

53 

53 

64 

29 

32 

73 

43 

54 

54 

38 

65 

OI 

33 

75 

44 

48 

55 

23 

66 

7-47 1 

j 






































CHAPTER XXX. 


GEODETIC CONSTANTS AND REDUCTION TABLES. 

290. Constants Depending on Spheroidal Figure of 
Earth.— The following are based on Clarke’s spheroid of 1886 : 

Equatorial semi-axis, a = 20926062. feet; 

“ “ a = 3963.3 miles; 

Polar “ b — 20855121. feet; 

“ “ b — 3949.8 miles; 

Equatorial radius, a — 6378206.4 meters; 

Polar “ b - 6356583.8 

Equatorial R. : Polar R. or a : b :: 294.98 : 293.98 ; 


log = 7.3206875; 
“ = 3-5980536; 
“ = 7.3192127; 

“ = 3 . 5965788 ; 

“ = 6.8046986; 
“ = 6.8032238; 
b_ _ 293.98 
a 294.98’ 


Circumference of equator = 24,901.96 miles. 

Area surface of earth = 196,940,400 square miles. 


a — 


lE ~ 


- = E 
2 

1 — e* 


Eccentricity 

Ellipticity 

• 9932313 ; 


.0067687 meters; 
a — b 1 

a 294.98’ 


a arc 1 


tt > 


log = 7.8305028; 
“ = 7.5294689; 
“ = 9-9970503; 
“ = 8.5097266. 


291. Numerical Constants. — Circumference of circle, 
diameter unity, 

= n — 3 -I 4 I 59 26 5 = log 0.4971499; 

2 7 T =6.2831853 = “ O.7981799; 

71 * = 9.8696044 = “ O.9942997. 


672 






NUMERICAL CONSTANTS . 


67 3 


Length of an arc, a, with radius, r, 


anr 

7 sC° = nearly 



+ ver sin 5 , 


c being the chord of the arc a. 


log. sine 1".= 4.6855748668; 

log. i sine 1".= 4.3845448711 ; 

a. c. log. sine 1".= 5.3144251 ; 

1" for radius = i mile 0.3072 inches; 
T “ “ “ = 18.431 “ 

nat. sine or tang. 1" = 0.00000485. 


Basis of natural logarithms. e = 2.7182818285 

Modulus of Briggs’s logarithms. m =0.4342944819 

Radius of the circle in seconds.r= 206264.8062 

“ “ “ “minutes.r= 3437*74677 

- « « * “ degrees. f- 57^957795 

Circumference of the circle in seconds. . . . 1296000 

‘ “ “ “ “ minutes. . .. 21600 

“ “ “ “ “ degrees. 360 


“ “ “ for the diameter = 1 

= 3.1415926536 

ASTRONOMICAL CONSTANTS (HARKNESS). 
Sidereal year = 365.2565578mean solar days. 

“ day = 23 h 56 m 4. s ioo mean solar time. 

Mean solar dav = 24 n 3“ 56. s 546 sidereal time. 

“ distance of the earth from the sun =92,800,000 miles. 


Log 

0.4342944819 
9-6377843ri3— 10 
5.314425133 2 

3- 5362738828 
1.7581226324 
6.1126050015 

4- 33445375 12 
2-5563025008 
0.0000000000 
o.497i49 8 727 


PHYSICAL CONSTANTS. 

Velocity of light (Harkness) = 186,337 miles per second = 299,878 km. per second. 
“ sound through dry air = ro9oV / 1 + 0.00367 t° C. feet per second. 


Table XLI. 

INTERCONVERSION OF ENGLISH LINEAR MEASURES, 


(From Smithsonian Geographical Tables.) 
Unit of linear measure is the yard. 


Inches. 

Feet. 

Y ards. 

Rods. 

Furlongs. 

Miles. 

1 

0.083 

0.028 

0.00505 

O.OOO12626 

0.0000157828 

12 

I. 

o -333 

0.06060 

O.OOI51515 

0.00018939 

36 

3 - 

1 . 

0.1818 

0.004545 

0.00056818 

198 

16.5 

5 • 5 

1 . 

O.025 

0.003125 

7920 

660. 

220. 

40. 

I. 

0.125 

63360 

5280. 

1760. 

320. 

8. 

1 . 



























674 GEODETIC CONSTANTS AND REDUCTION TABLES. 


i acre = 209 feet square.Error-j-i: 720 

1 “ =43,560 square feet. 

1 mile = 1760 yards =5280 feet =63,360 inches. 

To change log. miles to log. yards add 3.2455127; 

“ “ log. yards “ log. miles “ 6.7544873. 

Log. 3=0.4771212547; Log. 5280=3.7226339225; 

Log. 12=1.0791812460; Log. 1760=3.2455127. 

Other measures are the— 

Surveyor’s or Gunter’s chain = 4 rods = 66 feet = IOO 
links of 7.92 inches each. 

Fathom = 6 feet; Cable length = 120 fathoms. 

Hand = 4 inches; Palm = 3 inches; Span = 9 inches. 


Table XLII. 

INTERCONVERSION OF ENGLISH SQUARE MEASURES. 

(From Smithsonian Geographical Tables.) 

Unit of square measure is the square yard. 


Sq. Feet. 

Sq. Yards. 

Sq. Rods. 

Roods. 

Acres. 

Sq. 

Mi. 

I. 

0. mi 

0.00367309 

O.OOOO91827 

O.OOOO22957 


9 - 

1. 

0.0330579 

0.000826448 

0.000206612 


272.25 

30.25 

I. 

0.025 

O.OO625 


10890. 

1210. 

40. 

I. 

O.25 


43560. 

4840. 

160. 

4 - 

I. 


27878400. 

3097600. 

102400. 

2560. 

640. 

I. 


292. Length of the Meter in Inches. —According to 
various authorities 1 meter = in inches: 

39.370790 Kater, 1818. 

39.38092 Hassler, 1832. 

39.368505 Coast Survey, 1851-1858 (Hassler corrected). 

39.370432 Clarke, 1866-1873. 

39.36985 Lake Survey, 1885. 

39.3704316 Chief of Engineers, U. S. A., (letter) 1895. 

39-3777^6 Theoretic ten-millionth of quadrant (Clarke). 

39.37 By Act of Congress, 1866. 

39.37 U. S. Coast and Geodetic Survey, adopted 1891. 

















CONVERSION ENGLISH AND METRIC MEASURES. 6,75 

293. Interconversion of English and Metric Measures. 

—The units of measure of the two systems are the yard and 
the meter. The standard meter has its normal length at 
3 2 ° F. = 0° C.; the yard at —[— 62 0 F. Their relative values 
are 

I yard = of the meter. 


Table XLIII. 

TO CONVERT METRIC TO ENGLISH MEASURES. 

(From Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1893.) 


Meters to Inches. Meters to Feet. Meters to Yards. 

Kilometers. Miles. 

I = 39.3700 I = 3.28083 I = 1.093611 

I = 0.62137 

2= 78.74OO 2= 6.56167 2=2.187222 

2 = 1.24274 

3=Il8.IIOO 3= 9.84250 3=3.280833 

3 = 1.86411 

4=157.4800 4=13.12333 4=4.374444 

4 = 2.48548 

5 = 196.8500 5 = 16.40417 5 = 5.468056 

5 = 3.10685 

6=236.2200 6=19.68500 6=6.561667 

6 = 3.72822 

7=275.5900 7=22.96583 7=7.655278 

7 = 4-34959 

8 = 314.9600 8 = 26.24667 8 = 8.748889 

8 = 4.97096 

9 = 354-3300 9 = 29.52750 9=9.842500 

1---—--- 

9 = 5-59233 


Table XLIV. 

TO CONVERT ENGLISH TO METRIC MEASURES. 

(From Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1893.) 


Inches to Millimeters. Feet to Meters. 

I = 

25.4001 

1 = 0.304801 

2 = 

50.8001 

2 = 0.609601 

3 = 

76.2002 

3 = 0.914402 

4 = 

101.6002 

4 = 1.219202 

5 = 

127.0003 

5 = 1.524003 

6 = 

152.4003 

6 = 1.828804 

7 = 

177.8004 

7 = 2.133604 

• 8 = 

203.2004 

8 = 2.438405 

9 = 

228.6005 

9 = 2.743205 


Yards to Meters. 

Miles. Kilometers. 

I = 0.914402 

I = 1.60935 

2=1.828804 

2 = 3.21S69 

3 = 2.743205 

3 = 4.82804 

4 = 3.657607 

4 = 6.43739 

5 = 4.572009 

5 = 8.04674 

6 = 5.486411 

6 = 9.65608 

7 = 6.400813 

7 = n. '654^ 

8 = 7.315215 

8 = 12.86478 

9 = 8.229616 

9 = 14.48412 



















6/6 GEODETIC CONSTANTS AND REDUCTION TABLES. 

Table XLV. 

TO CONVERT METERS INTO STATUTE AND NAUTICAL MILES. 


Meter. 

Miles. 

Meter. 

Miles. 

I 

O.OOO62137 

6 

O.OO372822 

2 

O.OOI24274 

7 

O.OO434959 

3 

O.OO1864H 

8 

O 00497096 

4 

O.OO248548 

9 

O.OO559233 

5 

0.00310685 

10 

0.00621370 


Meters. 

Statute 

Miles. 

Nautical 

Miles. 

Meters. 

Statute 

Miles. 

Nautical 

Miles. 

Meters. 

Statute 

Miles. 

Nautical 

Miles. 

IO 

0.006 

0.005 

IOO 

0.062 

O.054 

1,000 

0.621 

O.540 

20 

0.012 

O.OII 

200 

O.124 

0.108 

2,000 

I.243 

I.079 

30 

0.019 

o.ol6 

300 

O. 186 

O. 162 

3.000 

1.864 

I.619 

40 

0.025 

0.022 

400 

O.249 

O. 216 

4,000 

2.486 

2.158 

50 

O.031 

0.027 

500 

O 311 

O.270 

5.000 

3.107 

2.698 

60 

O.037 

0.032 

600 

0-373 

O.324 

6,000 

3.728 

3-238 

70 

O.043 

0.038 

700 

0-435 

0.378 

7,000 

4-350 

3-777 

80 

0.050 

0.043 

800 

0 497 

O.432 

8,000 

4.971 

4-317 

90 

. 

0.056 

0.049 

900 

0 559 

0.486 

9,000 

5-592 

4-856 


294. Logarithms and Factors for Conversion of English 
and Metric Measures.— 


Table XLVI. 

LOGARITHMIC CONSTANTS FOR INTERCONVERSION OF 
METRIC AND COMMON MEASURES. 


To change log. of meters to log. of miles 
“ “ log. of meters to log. of yards 

“ log. of meters to log. of feet 
“ < ‘ log. of meters to log. of inches 

«« “ log. of miles to log. of meters 

“ ** log. of yards to log. of meters 

“ “ log. of feet to log. of meters 

“ “ log. of inches to log. of meters 


add 6.7933502 ; 
“ 0.0388629; 
“ 0.5159842; 
“ I-595I654; 

“ 3.2066498; 

“ 9 - 9 6 ii 37 1 ; 

“ 9-48401 5 8 ; 

“ 8.4048346. 

































METRIC AND ENGLISH EQUIVALENTS. 677 

Table XLVII. 

METRIC TO COMMON SYSTEM, WITH FACTORS AND 

LOGARITHMS. 


r- 

Units Compared. 

Logarithm 
of Factor. 

Reciprocal 
of Factor. 

Log. Rec. 
of Factor. 

Centimeters 

X 

0.3937 ~ inches. 

1-595165 

2-54 

0 404835 

Meters 

X 

3.2808333 = feet. 

0.515984 

0.304801 

1.4840158 

Kilometers 

X 

0.62137 — miles. 

9-79335 

1.60935 

0.20665 

Square centimeters X 

0.15500 — square inches. 

1 • I 9 ° 33 I 

6.45163 

0.809669 

Square meters 

X 

10.7639 — square feet. 

1.031968 

O.O929O34 

2.968032 

Hectares 

X 

2.47104 — acres. . 

0.39288 

0.404687 

1.60712 

Cubic centimeters 

X 

0.0610234 — cubic inches . 

8.785594 

16.3872 

1.214504 

Cubic meters 

X 

SS-SMS = cubic feet. 

1 -54795 

0.028317 

2.452047 

Cubic meters 

X 264.17 = U. S. gallons. 

2.421884 

0.0037854 

3.578116 


Millimeters X .03937 = inches. 

Millimeters -r- 25.4 — inches. 

Centimeters X -3937 — inches. 

Centimeters — 2.54 = inches. 

Meters X 39-37 = inches. (Act of Congress.) 
Meters X 3.281 = feet. 

Meters X 1.094 = yards. 

Kilometers X .621 = miles. 

Kilometers -f- 1.6093 = miles. 

Kilometers X 3280.8 = feet. 

Square millimeters X -0155 = square inches. 
Square millimeters 645.1 = square inches. 
Square centimeters X .155 = square inches. 
Square centimeters - 5 - 6.451 = square inches. 
Square meters X 10.764 = square feet. 

Square kilometers X 247.1 = acres. 

Hectares X 2.471 = acres. 

Hectares X 259 = square miles. 


Table XLVIII. 

MISCELLANEOUS METRIC EQUIVALENTS. 

1 millimeter = -^g inch.Error 4 - 1 : 62 

1 centimeter = f inch. ‘‘ +1:21 

1 meter = 3 feet 3! inches. “ + 1 : 8600 

1 kilometer = f mile. ^ ~ 1 : r 8o 

1 gram = 15-4 grains. “ — 1 : 4 ®° 

1 kilogram = 2 \ lbs. (avoirdupois). ‘‘ — 1 : 480 

1 liter = 1 quart. “ 1:18 

1 foot = T 3 o meters — 0.304801 meters 

1 fathom = 1 ■jyy meters = 1.829 

1 Gunter’s chain = 20y<j meters = 20.1168 


























PART VI. 


GEODETIC ASTRONOMY. 


CHAPTER XXXI. 

ASTRONOMIC METHODS. 

295. Method of Treatment —In the following pages only 
such outline of the subject is given as is indispensable as a 
guide to practical field operations. The mathematics of 
astronomy is extended and complex, and is omitted excepting 
the more practical working formulas, as volumes would be 
required for a complete exposition of this subject alone. 
The effort here has been to give directions for observing, 
examples of reduction and computation, and the essential 
field tables only. For more detailed information the student 
is referred to Doolittle’s or Chauvenet’s Practical Astronomies, 
Hayford’s Geodetic Astronomy, to the American Ephemeris, 
and to special tables and star catalogues. 

The arrangement of the following is similar to that of the 
rest of this book. The simpler and more approximate 
methods of determining azimuths, latitudes, and longitudes 
are given first, as they would be used in exploratory or rough 
geographic surveying. (Chap. IV.) Following these are given 
the more refined methods of determining the same quantities, 

67s 




GEODETIC ASTRONOMY. 679 

as initial positions for the extension of geodetic triangulation. 
(Art. 285.) 

296. Geodetic Astronomy. —The topographer should dis¬ 
tinguish in the beginning between an astronomic latitude and 
longitude and a geodetic latitude and longitude; in addition 
neither of these should be confused with celestial latitude and 
longitude. Astronomic latitudes and longitudes are referred 
to the action line of gravity at the station of observation. 
Geodetic latitudes and longitudes are referred to the gravity 
line which has been corrected for local deflection or station 
error. On the other hand celestial latitudes and longitudes 
refer to a system of spherical coordinates and, though much 
used by the astronomer, are rarely employed in topographic 
or geodetic operations. In geodetic astronomy the initial 
points of measurement are the equator and vernal equinox for 
the measure of declination and right ascension, whereas in 
celestial astronomy the ecliptic and vernal equinox furnish 
corresponding initial points. 

The field-work of the geodetic astronomer is of the most 
practical kind and has for its objects: 

1. To determine the astronomic latitude of the station; 

2. To determine the true local time at the instant of 
observation, or the true astronomic longitude of the station; 

3. To determine the azimuth of a line joining the observa¬ 
tion station with some other terrestrial point. 

Finally, as one of the operations performed in the above 
determinations consists in the finding of the horizon line, he 
consequently determines the zenith distance of some terres¬ 
trial object as a reference point for vertical triangulation. 
The zenith and a celestial object are therefore the two points 
on the celestial sphere always observed. The right ascension 
and declination of the object observed become known inde¬ 
pendently of the observations. 

297. Definitions of Astronomic Terms. —For some of 
the following definitions and explanations of the operations 


68 o 


ASTRONOMIC METHODS. 


of geodetic astronomy I am indebted to the admirable text¬ 
book on this subject recently published by Mr. John F. 
Hayford of the U. S. Coast and Geodetic Survey, to which 
the reader is referred for more detailed information upon this 
subject, as well as to Doolittle’s and Chauvenet’s Practical 
Astronomies. 

The astronomic latitude of a point on the surface of the 
earth is the angle between the line of action of gravity at that 
station and the plane of the equator. It is measured on the 
celestial sphere along the meridian from the equator to the 
zenith. 

The astronomic longitude of a point on the surface of the 
earth is the angle between the meridian plane of that point 
and some arbitrarily chosen meridian plane. The meridian 
of Greenwich, England, is accepted universally in the United 
States as this initial meridian for geodetic operations, and is 
generally accepted throughout the world in most nautical and 
geodetic work. The meridian of Washington, D. C., is 
sometimes used in the United States chiefly in connection 
with public land lines. 

Geodetic latitudes and longitudes are defined by applying 
the distinguishing explanation already made to the above 
definitions. 

In the operations of geodetic astronomy the bodies con¬ 
sidered are the sun, the moon, the stars, and the planets, 
including the earth and moon; also the satellites of the 
planets. As seen from the point of the observer these appear 
to move about within the range of vision, and their apparent 
motions appear quite complicated. To clearly orient himself 
upon the earth by observation upon these heavenly bodies 
he must have an accurate conception of their apparent 
motions. In the apparent motion of each of the heavenly 
bodies he sees not only its actual motion, but also the actual 
motion of the seemingly solid and movable earth upon which 
he stands, since both are in motion. As admirably expressed 


DEFINITIONS OF ASTRONOMIC TERMS. 65 i 

by Mr. Hayford, he is like a passenger upon a train at night 
looking out upon the moving lights of a town. He sees the 
lights apparently all in motion. In some cases the apparent 
motion of a light may be entirely due to his own motion. In 
other cases the lights upon which he looks may be those of a 
wagon or of another moving train, and their apparent motions 
are often due to their actual motions and those of himself. 

The horizon is the intersection with the celestial sphere of 
a plane passing through the eye of an observer perpendicular 
to the plumb-line or the line of action of gravity. The 
zenitJi is the point in which the action line of gravity produced 
upward intersects the celestial sphere, and is at right angles to 
the horizon of an observer, and opposite on the celestial 
sphere to the nadir. 

The plane of the equator is a plane of a great circle of the 
celestial sphere passing through the center of the earth and 
perpendicular to the axis of its rotation. The plane of the 
ecliptic is the plane of a great circle of the celestial sphere and 
is the plane of the orbit of the earth. The ecliptic itself is 
the intersection of the plane of the ecliptic with the celestial 
sphere. These two planes are the principal reference planes 
of astronomy . 

The eqninoxes are the two points in which the equator and 
ecliptic intersect each other, the angle of their intersection 
being about 23 0 27'. The vernal equinox is that at which the 
sun is found in the spring, and the autumnal that at which it is 
found in the fall. 

An hour-circle is the intersection of a plane passing through 
the axis of the earth with the celestial sphere, and all hour- 
circles are great circles passing through the poles. The hour- 
angle of a star is the angle measured along the equator 
between the meridian and the hour-circle passing through it. 

Right ascension of a celestial body is the angle measured 
along the equator between the hour-circles which pass through 
the star and the vernal equinox respectively. As right ascen- 


682 


ASTRONOMIC METHODS. 


sion is reckoned from west to east, opposite to the apparent 
motion of the stars, the sidereal time at the instant of a transit of 
a star is therefore the same as its right ascension. Right ascen¬ 
sion may also be expressed as the sidereal time elapsed between 
the passage of the vernal equinox and the star across the meri¬ 
dian. It is usually expressed in hours, minutes, and seconds. 

The declination of a celestial object is the angle between 
the line joining the center of the earth to the star or planet and 
the plane of the equator. It is also expressed as the angular 
distance of the heavenly body north or south of the equator, 
and is -f- when north and — when south. 

The culmination or transit of a celestial object across the 
meridian of the observer is the passage of that star across such 
meridian. As the meridian is a great circle, any star has two 
transits if considered for a complete revolution of the earth 
upon its axis; the first of these, called the upper transit or 
culmination, being that over the half of the meridian which 
includes the zenith. The second, called the lower transit or 
culmination, includes the transit over that half of the meridian 
which passes through the nadir. 

A sidereal day is the interval between two successive tran¬ 
sits of the vernal equinox across the same meridian. Sidereal 
time at the station of observation and at a fixed instant of 
time is the right ascension of the meridian, which is the same 
as the hour-angle of the vernal equinox counted in the direc¬ 
tion of the apparent motion of the stars. Sidereal time is 
zero hours, minutes, and seconds at the instant when the 
vernal equinox transits across the meridian. It includes 24 
hours, numbered consecutively from zero. 

An apparent solar day is the interval between two succes¬ 
sive transits of the sun across the meridian. The apparent 
solar time for any station of observer and any instant is the 
hour-angle of the real sun at that instant for that meridian. 

The mean solar day is the interval between successive 
transits of a fictitious mean sun over the same meridian. 


A S TRONOMIC NO TA TION . 


683 


Mean solar time , usually called mean time for any station of 
observer and instant, is the hour-angle of mean sun at that 
instant from that meridian. 

The standard time of any place is the mean solar time of 
the nearest fifteen degree meridian of longitude west of 
Greenwich. To reduce local mean solar time to standard 
time apply as a correction the difference of longitude of the 
place and its standard meridian. 

Idle equation of time is the correction to be applied to the 
apparent time to reduce it to mean time. It is given in the 
American Ephemeris. 

The civil day commences and ends at midnight. Its hours 
are counted from zero to 12 between midnight and noon, and 
from zero to 12 between noon and midnight. Th z astronomic 
day commences at noon of the civil day of the same date, 
and its hours are numbered from zero to 
24, from noon of one day to noon of the 
next. Civil time is local mean solar time 
based on the civil day. 

298. Astronomic Notation. — The 
following is the notation employed in 
astronomic formulas and computations: 

T = civil time at any place; 

T s = sidereal time corresponding to T 
— right ascension of the meri¬ 
dian of the place; 

T, n = astronomic mean time correspond¬ 
ing to T s ; 

I = interval of mean solar time; 

C s — correction to convert interval I 

into mean time (Ephemeris, Table III); 

I' — interval of sidereal time corresponding to /; 
a s = R. A. mean sun for next preceding mean noon for 
place, P, and date, D t , 

— sidereal time of mean noon for place and date; 


N 



Fig. 179. —Latitude, 
Declination, and 
Altitude. 



684 


ASTRONOMIC METHODS. 


E — equinox; 

t — hour-angle = difference between sidereal and mean 
time; 

A — azimuth of star or other celestial object; 
h — altitude of same; 

a = R. A. = right ascension of celestial object; 

0 = latitude of place = angle of pole above horizon; 
d = declination of celestial object; 
z = its observed zenith distance — 90° — Jr, 
z m — its observed meridional zenith distance; 

C = its true meridional zenith distance ; 
p = its polar distance = 90° — d; 

q — its parallactic angle or angle at star between pole and 
zenith. 


299. Fundamental Astronomic Formulas. —To find alti¬ 
tude, given latitude, declination, and hour-angle : 

sin h — sin 0 sin 6 -f- cos 0 cos d cos t. . . (96) 

By means of logarithms and an auxiliary angle M, where 


sin d = m sin M and cos d cos t ~ m cos M, 


we have 


sin h = m cos (0 — M). 


(97) 


To find azimuth, given latitude, declination, and hour-angle : 

.(98) 


tan t cos M 

tan A — . , , /17"\* 

sin (0 — M ) 


and 


tan A — — 


sin t 


cos 0 tan d(i — tan 0 cos d cos t)' * ^ 9 ) 


Where a series of observations are made, this formula is sim¬ 
plified into the following working form: 






FUNDAMENTAL ASTRONOMIC FORMULAS . 685 


tan A — 


a sin t 
1 — b cos t' 


where a =: sec 0 cot 3 , and b — tan 0 cot 3 . 

To find declination , given altitude and azimuth : 

sin 3 — sin 0 sin h — cos 0 cos h cos A ; . (100) 

or, for logarithmic computation, 


sin 3 = m sin (0 — M ); .... (101) 

or, having t , 

tan 6 = tan (0 — M) cos t. 

To find hour-angle, given altitude and azimuth : 


tan A sin M 

tan t = - — ..( I02 ) 


cos (0 — M)' 


To find hour-angle and azimuth in terms of zenith distance : 


cos z — sin 0 sin 3 

cos t — ---r-; 

COS 0 cos o 


(103) 


cos A = 


sin 0 cos z — sin 3 
cos 0 sin z 


• • (104) 


To find hour-angle and zenith distance of a star at elonga¬ 


tion 


tan 0 

cos t — - -- ; 

tan 0 

. (105) 

• 

. cos 3 
sin A — 

COS 0 

. (106) 

sin 0 
cos z = -— 
sin 0 

. (107) 









686 


ASTRONOMIC METHODS. 


To find hour-angle and azimuth of a star in the horizon or 
at time of rising or setting: 


cos t — 


cos A = 


— tan 0 tan 6 ; 

sin S 
cos 0 


(i°8) 

( I0 9 ) 


To find hour-angle, zenith distance, and parallactic angle 
for transit of a star across prime vertical: 

tan S 

cos t = -:.(no) 

tan 0 

sin 3 ( x 

cos z — — --;.(i 11) 

sin 0 

COS 0 , * 

sin q =...(112) 

cos d 

300. Finding the Stars.—The following brief statement 
of the positions of the more prominent stars as referred, one 
to the other, is derived from Lieutenant Qualtrough’s “Sail¬ 
or’s Manual.” The most conspicuous stars have been desig¬ 
nated by names, and the stars in each constellation are dis¬ 
tinguished, for reference, by letters and numbers. The letters 
used for this purpose are the small letters of the Greek alpha¬ 
bet. 

In finding any star in the heavens, it is customary to refer 
to some one star or constellation as known : The Great Bear, 
called also by the Latin name of Ursa Major , in the northern 
part of the heavens and consisting of seven principal stars, is 
the most convenient for the purpose. 

The two stars a and /? point nearly to Polaris , or the Pole 
Star, and are hence called the Pointers. 

A line from Polaris through 7/, the last of the tail, passes 
at 31 0 beyond 7/ through Arc turns, a very bright star. 







FINDING THE STARS. 


687 


A line from Polaris perpendicular to the line of the 
Pointers and on the opposite side to the Great Bear passes at 
48° distance through Capella , one of the brightest stars. 

In the same line, about the same distance on the opposite 
side of the Pole, is a Lyrce , also called Vega and Lyra, a 
large white star in the Harp. 

At one third of the distance from Arcturus to a Lyrae is 
Alphacca, the brightest star of a semicircular group called the 
Northern Crown . 

About 23 0 to the eastward of a Lyrae, and about the same 
distance as this star is from Polaris, is f-i Cygni , the bright 
star in the Swan. 

A line from Polaris passing between this last and ol Lyrae, 
and produced to an equal distance between them, passes 
through a Aquilce y or Altair, a bright star between two small 
ones. 

A line from Polaris drawn between Capella and a star close 
to the eastward of it passes to the westward of the constella¬ 
tion Orion. The two northern stars of the four at the corners 
are the shoulders, the northernmost of which is a Orionis . 
The brightest of the two southern stars, the feet, is called 
Rigel. In the middle are three stars forming the Belt , the 
northernmost of which is nearly on the equator. 

About 25 0 to the northwestward of the Belt, and not far 
out of its line, is Aldebaran, which may be known by its red 
color. 

A line from Aldebaran through the Belt passes at about 
20° on the other side through Sirius , the brighest star in the 
heavens. 

Sirius , the eastern shoulder, and Procyon , to the eastward 
of Orion and northward of Sirius, form an equilateral tri¬ 
angle. 

Midway between the Great Bear and Orion are the Twins, 
Castor and Pollux , the latter the southern and brighter, about 


688 


ASTRONOMIC METHODS. 


4° apart. The line from Polaris to Procyon passes between 
them. 

A line from Rigel through Procyon passes at an equal dis¬ 
tance beyond to the northward of Regulus. 

A line from Polaris through Ursa Majoris passes at yo° 
distance through Spica Virginis. 

A line from Regulus through Spica passes at 45 0 distance 
through Ant ares, a bright reddish star. 

The line from the Pointers carried through the Pole to 
about 75 0 beyond it, passes through Marcab or a Pegasi. 

A line from Polaris through Marcab passes at 45 0 distance 
through Fomalhaut , a very bright star. 

Achernar, Fomalhaut , and Canopas are in a line and nearly 
equidistant, being about 40° apart. 

The Southern Cross is about as far from the South Pole as 
the Great Bear is from the North Pole —y is the head, and a 
the foot. 

When some stars are known, the rest are easily found by 
the times of their meridian passages and their declinations. 
A star may also be identified by means of its altitude or 
azimuth, computed approximately. 

301. Parallax.— The word parallax is generally used to 
designate the apparent displacement due to the change in the 
position of the observer. As used in referring to the sun or 
other celestial object, it is employed to indicate the difference 
of direction of such object as seen from the center of the earth 
and from a station on the surface of the earth. The hori¬ 
zontal parallax , or that of the object in the horizon of the 
observer, is the angle subtended at the sun by the radius of 
the earth. 

The horizontal parallax may be represented by the 
formula 


P = 


2 S i n ! " = 9 " (approximate), 


(ii 3 ) 




PARALLAX AND REFRACTION . 689 

in which P = horizontal parallax of the sun in seconds of arc; 
r = radius of the earth; and 

d — distance between the center of the earth and 
the sun. 

If it is desired to know the exact value of the equatorial 
parallax , which is the parallax of a celestial object as seen by 
an observer at the equator, this may be found in the American 
Ephemeris. 

The parallax of the sun at any position above the horizon 
may be determined by the formula 

p — P cos A } .(114) 

in which p — parallax of the sun at any position, and 

A — angle at the earth’s surface between the object 
observed and the horizon. 

The following table (XLIX), from Hayford, gives the 
parallax of the sun for any date and altitude. As the distance 
of the sun is nearly the same for the same date in different 
years, this table may be used for any year. 

302. Refraction —A ray of light from any celestial object 
encounters, as it approaches the earth, successive strata of air, 
each more dense than the upper. In passing through these 
the ray is continually bent out of the straight line so as to 
cause the portion of its path through the atmosphere of the 
earth to be a curve. This is refraction. 

Refraction acts according to the following general laws : 

1. When a ray passes from a. lighter to a denser medium it 
is refracted towards the normal to the separating surface by an 
amount which is a function of the angle between the ray and the 
normal , and of the densities of the tzvo media. 

2. A plane containing a normal and an original ray also 
contains a refracted ray. 

The effect of refraction is noted directly in measuring 
altitudes, refraction always making the observed altitude too 


690 


ASTRONOMIC METHODS. 


great (Arts. 166 and 239). Refraction has no apparent effect 
on azimuth. As the theory and computations of refraction 
are complicated, the topographer is referred for the application 
of the effects of refraction to the following tables, derived 
from Hayford. 

Table L gives the mean refraction, R m , under a barometric 
pressure of 29.9 inches and temperature of 50 degrees Fahren¬ 
heit. As mean refraction is a function of the altitude, it must 
be multiplied by a factor, C B , derived from Table LI, if the 
barometric reading is not 29.9 inches. Finally, the mean 
refraction must be multiplied by the factor C D (Table LII) 
where the temperature of the observation station differs from 
50 degrees. Refraction R, as computed by these tables, is 

R = R m C B C D C A .(115) 


PARALLAX OF SUN. 


69I 


Table XLIX. 

PARALLAX OF SUN (/) FOR FIRST DAY OF EACH MONTH. 

(From Hayford’s Geodetic Astronomy.) 


1 - n 

Altitude. 

-, 

Jan. 1st. 

Feb. 1st. 
Dec. 1st. 

Mar. 1st. 
Nov. 1st. 

• 

April 1st. 

Oct. 1st. 

May 1st. 

Sept. 1st. 

June 1st. 

Aug. 1st. 

July 1st. 

0° 

9 ".o 

g" .0 

8". 9 

8". 9 

8". 8 

8". 7 

8". 7 

3 

9 .0 

9 .0 

8 .9 

8 .8 

8 .8 

8 .7 

8 -7 

6 

9 .0 

8 .9 

8 .9 

8 .8 

8 -7 

8 -7 

8 .7 

9 

8 .9 

8 .9 

8 .8 

8 .8 

8 -7 

8 .6 

8 .6 

12 

8 .8 

8 .8 

8 -7 

8 -7 

8 .6 

8 .5 

8 -5 

15 

8 .7 

8 *7 

8 .6 

8 .6 

8 .5 

8 .4 

8 *4 

18 

8 .6 

8 .6 

8 .5 

8 .4 

8 .4 

8 -3 

8 .3 

21 

8 .4 

8 .4 

8 -3 

8 .3 

8 .2 

8 .2 

8 .1 

24 

8 .2 

8 .2 

8 .2 

8 .1 

8 .0 

8 .0 

8 .0 

27 

8 .0 

8 .0 

8 .0 

7 -9 

7 *8 

7 .8 

7 .8 

30 

7 -8 

7 *8 

7 -7 

7 -7 

7 .6 

7 *6 

7 -6 

33 

7 -6 

7 -5 

7 -5 

7 *4 

7 *4 

7 *3 

7 -3 

36 

7 -3 

7 -3 

7 • 2 

7 -2 

7 -i 

7 -i 

7 -o 

39 

7 -o 

7 -o 

6 .9 

6 .9 

6 .8 

6 .8 

6 .8 

42 

6 .7 

6 .7 

6 .6 

6 .6 

6 .5 

6 .5 

6 .5 

44 

6 -5 

6 *5 

6 .4 

6 .4 

6 -3 

6 *3 

6 -3 

46 

6 -3 

6 .2 

6 .2 

6 .2 

6 .1 

6 .1 

6 .0 

48 

6 .0 

6 .0 

6 .0 

5 -9 

5 -9 

5 -8 

5 *8 

50 

5 -8 

5 .8 

5 -7 

5 -7 

5 -6 

5 -6 

5 -6 

52 

5 -6 

5 -5 

5 -5 

5 -4 

5 -4 

5 -4 

5 -4 

54 

5 -3 

5 -3 

5 .2 

5 -2 

5 -2 

5 -i 

5 -i 

56 

5 .0 

5 -o 

5 -o 

5 .0 

4 -9 

4 -9 

4 -9 

53 

4 .8 

4 .8 

4 -7 

4 -7 

4 -7 

4 .6 

4 .6 

60 

4 -5 

4 • 5 

4 -5 

4 -4 

4 -4 

4 -4 

4 -4 

62 

4 .2 

4 .2 

4 .2 

4 .2 

4 -i 

4 -i 

4 -i 

64 

4 .0 

3 -9 

3 -9 

3 -9 

3 -8 

3 -8 

3 -8 

66 

3 -7 

3 -7 

3 * 6 

3 -6 

3 -6 

3 -6 

3 -5 

68 

3 -4 

3 -4 

3 -4 

3 -3 

3 -3 

3 -3 

3 *3 

70 

3 -i 

3 -i 

3 -i 

3 -o 

3 -o 

3 -o 

3 -o 

72 

2 .8 

2 .8 

2 .8 

2 .7 

2 -7 

2 .7 

2 -7 

74 

2 .5 

2 .5 

2 .5 

2 .4 

2 .4 

2 .4 

2 .4 

76 

2 .2 

2 .2 

2 .2 

2 .1 

2 .1 

2 .1 

2 .1 

78 

1 .9 

I .9 

1 .9 

1 .8 

1 .8 

1 .8 

1 .8 

80 

1 .6 

i .6 

1 .6 

1 -5 

1 -5 

1 -5 

1 .5 

82 

1 .2 

1 .2 

I .2 

1 .2 

1 .2 

1 .2 

1 .2 

84 

0 .9 

0 .9 

0 .9 

0 .9 

0 .9 

0 .9 

0 .9 

86 

0 .6 

0 .6 

0 .6 

0 .6 

0 .6 

0 .6 

0 .6 

88 

0 .3 

0 -3 

0 -3 

0 -3 

0 -3 

0 .3 

0 .3 

90 

0 .0 

0 .0 

0 .0 

0 .0 

0 .0 

0 .0 

0 .0 


mhimw h » m to m n 00 u u u co 4- -t* -P* 4 *muun O' 0 O' O'^i -*i vi 00 00 000 Zenith 

O to 4 * o> co o 14 4- O' c« o to 4- O' 00O 04 * O' co o to 4- O' coh^m O c>» 0"0 to cn cdmo-vi o Distance. 






























692 


A STKONOMIC ME THODS 


Table L. 

MEAN REFRACTION (Em). BAROMETER 29.9 INCHES, TEMP. 

50° F. 

(From Hayford’s Geodetic Astronomy.) 


Altitude. 

Mean 

Refraction. 

Change per 
Minute. 

Altitude. 

Mean 

Refraction. 

Change per 

Minute. 

Altitude. 

Mean 

Refraction. 

Change per 

Minute. 

Altitude. 

Mean 

Refraction. 

Change per 

Minute. 

O 

0 00' 

34 

08". 6 

n".66 

7 

°oo / 

y 

24".2 

o'' 

•95 

!9 

0 oo' 

2 f 

47" 

.6 

0". 16 

33 

0 oo' 

l' 

29". 4 

o".o6 


IO 

3 2 

*5 -9 

10 .88 


IO 

7 

14 .9 

O 

• 9 i 


20 

2 

44 

.6 

0 .15 


20 

I 

28 .2 

0 .06 


20 

30 

3 1 -i 

IO . IO 


20 

7 

06 .0 

O 

.88 


40 

2 

41 

.6 

0 .15 


40 

I 

27 . I 

0 .05 


30 

28 

53 -9 

9 * 6 4 


3 ° 

6 

57 -4 

O 

• 84 

20 

OO 

2 

38 

•7 

O . 14 

34 

OO 

I 

26 .1 

0 .05 


40 

27 

18 .2 

9 .20 


40 

6 

49 -i 

O 

.81 


20 

2 

35 

•9 

0 .14 


20 

1 

25 .0 

0 .05 


50 

25 

49 -8 

8 .50 


50 

6 

41 .2 

O 

.78 


40 

2 

33 

.2 

0 .13 


40 

I 

24 .O 

0 .05 

I 

OO 

24 

28 .3 

7 .82 

8 

OO 

6 

33 -5 

O 

.76 

21 

OO 

2 

30 

.6 

0 .13 

35 

OO 

I 

23 .0 

0 .05 


IO 

23 

*3 -5 

7 -17 


IO 

6 

26 .0 

O 

•73 


20 

2 

28 

• I 

0 .13 


20 

I 

22 .O 

0 .05 


20 

22 

04 .9 

6 .58 


20 

6 

18 .9 

O 

.70 


40 

2 

25 

.6 

0 .12 


40 

I 

21 .0 

0 .05 


30 

21 

01 .8 

6 .06 


3 ° 

6 

12 .0 

O 

.68 

22 

OO 

2 

23 

.2 

O . 12 

36 

OO 

I 

20 .0 

0 .05 


40 

20 

°3 -7 

5 «6o 


40 

6 

°5 *3 

O 

.66 


20 

2 

20 

•9 

0 .12 


30 

I 

18 .5 

0 .05 


50 

19 

09 .8 

5 -20 


50 

5 

58 -9 

O 

•63 


40 

2 

18 

.6 

0 .11 

37 

00 

I 

17 .1 

0 .04 

2 

OO 

18 

19 .7 

4 *84 

9 

OO 

5 

52 .7 

O 

.61 

23 

OO 

2 

l6 

•4 

0 .11 


30 

I 

15 .7 

O .04 


IO 

17 

33 

4 .50 


20 

5 

40 .8 

O 

•58 


20 

2 

14 

.2 

0 .11 

38 

OO 

I 

14 .4 

0 .04 


20 

l6 

49 .7 

4 .18 


40 

5 

29 .7 

O 

•54 


40 

2 

12 

. 1 

0 .10 


30 

1 

13 -i 

0 .04 


30 

16 

09 -5 

3 .88 

IO 

OO 

5 

19 .2 

O 

• 5 i 

24 

OO 

2 

IO 

. 1 

O .IO 

39 

00 

I 

11 .8 

0 .04 


40 

15 

32 .1 

3 -62 


20 

5 

09 .4 

O 

.48 


20 

2 

08 

T 

O .IO 


30 

I 

10 .5 

O .04 


50 

14 

57 -i 

3 -39 


40 

5 

OO . I 

O 

46 


40 

2 

06 

.1 

O . IO 

40 

OO 

I 

09 -3 

0 .04 

3 

OO 

14 

24 -3 

3 .18 

II 

OO 

4 

51 .2 

O 

43 

25 

00 

2 

04 

.2 

O .09 


3 ° 

I 

08 .1 

0 .04 


10 

!3 

53 -6 

2 .98 


20 

4 

42 .8 

O 

40 


20 

2 

02 

• 4 

0 .09 

4 i 

OO 

I 

06 .9 

0 .04 


20 

1 3 

24 .8 

2 -79 


40 

4 

35 .0 

0 

38 


40 

2 

OO 

.6 

0 .09 


30 

I 

05 -7 

0 .04 


30 

12 

57 -8 

2 .61 

12 

OO 

4 

27 -5 

O 

37 

26 

OO 

I 

58 

8 

O .09 

42 

OO 

I 

04 .6 

0 .04 


40 

12 

32 .5 

2 .46 


20 

4 

20 .3 

0 

35 


20 

I 

57 

I 

O .09 


30 

I 

03 -5 

0 .04 


50 

12 

08 .7 

2 -33 


40 

4 

*3 -5 

O 

33 


40 

I 

55 

4 

0 .08 

43 

OO 

I 

02 .4 

0 .04 

4 

OO 

II 

46 .0 

2 .20 

13 

OO 

4 

07 .1 

0 

32 

27 

OO 

I 

53 

8 

0 .08 


30 

I 

01 .3 

0 .04 


IO 

II 

24 .6 

2 .09 


20 

4 

00 .9 

O 

30 


20 

I 

52 

2 

0 .08 

44 

OO 

I 

OO .2 

0 .03 


20 

II 

04 .2 

1 .98 


40 

3 

55 -i 

O 

28 


40 

I 

50 

6 

0 .08 


30 

I 

59 -2 

0 .03 


30 

IO 

44 -9 

00 

00 

H 

14 

OO 

3 

49 -5 

O 

27 

28 

OO 

I 

49 

1 

0 .08 

45 

OO 

O 

58 .2 

0 .03 


40 

IO 

26 .5 

1 .79 


20 

3 

44 .2 

O 

2b 


20 

I 

47 

6 

0 .07 


30 

O 

57 -2 

0 .03 


50 

IO 

09 . I 

1 .70 


40 

3 

39 - 1 

O 

25 


40 

I 

46 

1 

0 .07 

46 

00 

O 

56 .2 

0 .03 

5 

OO 

9 

52 .6 

1 .61 

15 

OO 

3 

34 .1 

O 

24 

29 

OO 

I 

44 

6 

0 .07 


3 ° 

O 

55 .2 

0 .03 


IO 

9 

3 6 -9 

1 -54 


20 

3 

29 -4 

O 

23 


20 

I 

43 

2 

O .07 

47 

OO 

O 

54 -2 

o .03 


20 

9 

21 .9 

1 .46 


40 

3 

24 .8 

O 

23 


40 

I 

41 

8 

0 .07 


30 

O 

53 -3 

0 .03 


3 ° 

9 

07 .6 

1 .40 

l6 

OO 

3 

20 .4 

O 

22 

3 ° 

OO 

I 

40 

5 

O .07 

48 

OO 

O 

52 .5 

0 .03 


40 

8 

54 -o 

i *33 


20 

3 

l6 . I 

O 

21 


20 

I 

39 

I 

O .07 


30 

O 

51 .6 

0 .03 


50 

8 

41 .0 

I .27 


40 

3 

12 .0 

O 

20 


40 

I 

37 

8 

0 .06 

49 

00 

O 

50 -7 

0 .03 

6 

OO 

8 

28 .6 

I .22 

17 

OO 

3 

08 .2 

O 

>9 

3 i 

OO 

I 

36 

6 

0 .06 


3 ° 

O 

49 -8 

0 .03 


IO 

8 

16 .7 

I . l6 


20 

3 

04 .5 

O 

i 9 


20 

I 

35 

3 

0 .06 

5 ° 

OO 

O 

48 .9 

0 .03 


20 

8 

°5 -3 

I . 12 


40 

3 

OO .9 

O 

l8 


40 

I 

34 

1 

0 06 


30 

O 

48 .0 

0 .03 


30 

7 

54 -3 

i .07 

18 

OO 

2 

57 -4 

O 

17 

32 

OO 

I 

32 

9 

0 .06 

5 i 

OO 

O 

47 -2 

0 .03 


40 

7 

43 -9 

I .02 


20 

2 

54 

O 

17 


20 

I 

31 

8 

0 .06 


30 

O 

46 .3 

0 .03 

— 

50 

7 

33 -9 

0 .98 


40 

2 

50 -7 

O 

l6 


40 

[ 

30 

6 

0 .06 

52 

OO 

O 

45 *5 

0 .03 





















































CORRECTION TO MEAN RETRACTION. 


°93 


Table LI. 


MEAN REFRAC¬ 
TION ( R M ). — Cont. 


Altitude. 

Mean 

Refraction. 

Change per 
Minute. 

___— 

52 ° 

3 °' 

o' 

44" 

•7 

o".03 

53 

OO 

0 

43 

•9 

0 .03 


30 

0 

43 

.1 

0 .03 

54 

OO 

0 

42 

•3 

0 .03 


30 

0 

41 

.6 

0 .03 

55 

OO 

0 

40 

.8 

0 .03 


30 

0 

40 

.0 

0 .03 

5*5 

OO 

0 

39 

•3 

O .025 

57 

OO 

0 

37 

.8 

O .024 

58 

OO 

0 

36 

•4 

O .023 

59 

OO 

0 

35 

.O 

O .023 

60 

OO 

0 

33 

.6 

O .022 

. 61 

OO 

0 

32 

•3 

O .022 

62 

OO 

0 

31 

.0 

O .022 

63 

OO 

0 

29 

•7 

O .022 

64 

OO 

0 

28 

•4 

0 .021 

65 

OO 

0 

27 

.2 

O .021 

66 

OO 

0 

25 

•9 

O .021 

67 

OO 

0 

24 

•7 

O .020 

68 

OO 

0 

23 

.6 

0 .020 

69 

OO 

0 

22 

•4 

O .020 

70 

OO 

0 

21 

.2 

O .Oig 

71 

OO 

0 

20 

. I 

O .OI9 

72 

OO 

0 

18 

•9 

0 .019 

73 

OO 

0 

17 

.8 

0 .018 

74 

OO 

0 

16 

•7 

0 .018 

75 

OO 

0 

15 

.6 

0 .018 

76 

OO 

0 

14 

•5 

0 .018 

77 

OO 

0 

T 3 

•5 

0 .018 

78 

OO 

0 

12 

•4 

0 .018 

79 

OO 

0 

11 

• 3 

0 .018 

80 

OO 

0 

10 

• 3 

0 .018 

81 

OO 

0 

09 

.2 

0 .018 

82 

OO 

0 

08 

.2 

0 .018 

83 

OO 

0 

07 

.2 

0 .018 

84 

OO 

0 

06 

. 1 

0 .018 

85 

OO 

0 

05 

. I 

0 .018 

86 

OO 

0 

04 

. 1 

O .OI7 

87 

OO 

0 

03 

. I 

0 .017 

88 

OO 

0 

02 

.O 

0 .017 

89 

OO 

0 

01 

.O 

O .017 

90 

OO 

0 

00 

.O 

0 .017 






_ 


CORRECTON (Cb) TO MEAN REFRAC¬ 
TION DEPENDING UPON READING OF 
BAROMETER. 

R = (Rm)(Cb)(Cd)(Ca)- 


(From Hayford’s Geodetic Astronomy.) 


-—- 

Barometer. 

Inches. 

Barometer. 

Millimeters. 

C B 

Barometer. 

Inches. 

Barometer. 

Millimeters. 

G 5 

1 

Barometer. 

Inches. 

Barometer. 

Millimeters. 

C* 

20.0 

508 

0.670 

24.2 

615 

0.809 

28.4 

721 

0.949 

20.1 

511 

0.673 

24-3 

617 

0.813 

28.5 

724 

0-953 

20.2 

5*3 

0.676 

24.4 

620 

0.816 

28.6 

726 

0.956 

20.3 

516 

0.679 

24.5 

622 

0.820 

28.7 

729 

0-959 

20.4 

518 

0.682 

24.6 

625 

0.823 

28.8 

732 

0.963 

20.5 

521 

0.685 

24.7 

627 

0.826 

28.9 

734 

0.966 

20.6 

523 

0.688 

24.8 

630 

0.829 

29.0 

737 

0.970 

20.7 

526 

0.692 

24.9 

632 

0.832 

29. I 

739 

0-973 

20.8 

528 

0.696 

25.0 

635 

0.835 

29.2 

742 

0.976 

20.9 

53 1 

0.699 

25.1 

637 

0.838 

29-3 

744 

0.979 

21 .O 

533 

0.703 

25.2 

640 

0.842 

29.4 

747 

0.983 

21.1 

536 

0.706 

25-3 

643 

0.846 

29-5 

749 

0.986 

21.2 

538 

0.709 

25.4 

645 

0.849 

29.6 

752 

0.989 

21.3 

54 i 

0.712 

25.5 

648 

0.853 

29.7 

754 

O.992 

21.4 

544 

0.7x6 

25.6 

650 

0.856 

29.8 

757 

0.996 

21.5 

546 

0.719 

25-7 

653 

0.859 

29.9 

759 

0.999 

21.6 

549 

0.722 

25.8 

655 

0.862 

30.0 

762 

1.003 

21.7 

55 1 

0.725 

25-9 

658 

0.866 

30.1 

765 

1.007 

21.8 

554 

0.729 

26.0 

660 

0.869 

30.2 

767 

I .OIO 

21.9 

556 

0.732 

26. i 

663 

0.872 

30-3 

770 

1.013 

22.0 

559 

0-735 

26.2 

665 

0.875 

30.4 

772 

1.0x6 

22.1 

56 r 

0-739 

26.3 

668 

0.879 

30-5 

775 

I .020 

22.2 

564 

0.742 

26.4 

671 

0.882 

30.6 

777 

1.023 

22.3 

566 

0.746 

26.5 

673 

0.885 

30-7 

780 

1.026 

22.4 

569 

0.749 

26.6 

676 

0.889 

30.8 

782 

1.029 

22.5 

572 

0.752 

26.7 

678 

0.892 

30-9 

785 

1.033 

22.6 

574 

0-755 

26.8 

681 

0.896 

3r.o 

787 

1.036 

22.7 

576 

0-759 

26.9 

683 

0.899 




22.8 

579 

0.762 

27.0 

686 

0.902 




22.9 

582 

0.766 

27.I 

688 

0.905 




23.0 

584 

0.770 

27.2 

691 

0.909 




23.1 

587 

0-773 

27-3 

693 

0.912 




23.2 

589 

0.776 

27.4 

696 

0.916 




23-3 

592 

0.779 

27-5 

690 

0.920 




23-4 

594 

0.783 

27.6 

701 

0.923 




23-5 

597 

0.786 

27.7 

704 

0.926 




23.6 

599 

0.789 

27.8 

706 

0.929 




23.7 

602 

0.792 

27.9 

7°9 

0-933 




23.8 

605 

0.796 

28.0 

7 11 

0.936 




23.9 

607 

0-799 

28.1 

7 r 4 

0.939 




24.0 

610 

0.803 

28.2 

716 

0.942 




24 . I 

612 

0.806 

28.3 

719 

0.946 





































694 


A S TRONOMIC ME THODS. 


Table LII. 


CORRECTION (Cd) TO MEAN REFRACTION DEPENDING UPON 
READING OF DETACHED THERMOMETER. 


R = (Rm)(Cb)(C d )(Ca). 

(From Hayford’s Geodetic Astronomy.) 


| Temp. 
Fahr. 

Temp. 

Cent. 

C D 

Temp. 

Fahr. 

Temp. 

Cent. 

C D 

— 

Temp. 

Fahr. 

Temp. 

Cent. 

C D 

— 25 ° 

- 3 i °-7 

I . 172 

20° 

-6°.7 

1.062 

65° 

18°. 3 

0.972 

-24 

-31 .1 

1.169 

21 

— 6 .1 

1.060 

66 

18 .9 

0.970 

-23 

— 30 .6 

1.166 

22 

-5 -6 

1.058 

67 

19 .4 

0.968 

— 22 

-30 .0 

1.164 

23 

-5 .0 

1.056 

68 

20 .O 

0.966 

— 21 

-29 -4 

1.161 

24 

-4 .4 

1.054 

69 

20 .6 

0.964 

— 20 

— 28 .9 

1.158 

25 

-3 -9 

1-051 

70 

21 .1 

0.962 

-19 

—28 .3 

1.156 

26 

-3 -3 

1.049 

71 

21 .7 

0.961 

— l8 

— 27 .8 

*•153 

27 

-a .8 

1.047 

72 

22 .2 

0.959 

— '7 

—27 .2 


28 

— 2 .2 

1.045 

73 

22 .8 

0.957 

— l6 

—26 .7 

1.148 

29 

-t .7 

1.043 

74 

23 -3 

o -955 

-*5 

— 26 .1 

1-145 

30 

— 1 .1 

1.041 

75 

23 -9 

0-953 

— 14 

-25 .6 

1-143 

31 

—0 .6 

1.039 

76 

24 -4 

0.952 

— J 3 

—25 .0 

1 140' 

32 

0 .0 

1.036 

77 

25 .0 

0.950 

— 12 

-24 .4 

1.138 

33 

+0 .6 

1.034 

78 

25 -6 

0.948 

— II 

-23 .9 

1*135 

34 

I . I 

1.032 

79 

26 .1 

0.946 

— IO 

-23 -3 

i-i 33 

35 

1 -7 

1.030 

80 

26 .7 

0-945 

- 9 

—22 .8 

1-130 

36 

2 .2 

1.028 

81 

27 .2 

0-943 

- 8 

— 22 .2 

1.128 

37 

2 .8 

1.026 

82 

27 .8 

0.941 

- 7 

—21 .7 

1-125 

38 

3 -3 

I .024 

83 

28 .3 

0-939 

— 6 

— 21 .1 

1.123 

39 

3 -9 

T .022 

84 

28 .9 

0.938 

— 5 

—20 .6 

1.120 

40 

4 -4 

I .020 

85 

29 -4 

0.936 

- 4 

— 20 .O 

M 

M 

H 

00 

41 

5 -o 

1.018 

86 

30 .0 

0-934 

“ 3 

—19 .4 

1-115 

42 

5 -6 

1.016 

87 

30 .6 

0-933 

— 2 

— 18 .9 

1.113 

43 

6 .1 

I .OI4 

88 

31 -i 

0.931 

— I 

-18 .3 

I. Ill 

44 

6 .7 

I .012 

89 

3 i -7 

0.929 

O 

— 17 .8 

1.108 

45 

7 -2 

I .OIO 

90 

32 .2 

0 928 

4 - 1 

— 17 .2 

1.106 

46 

7 -8 

1.00S 

9 i 

3 2 -8 

0.926 

2 

-16 .7 

1.103 

47 

8 -3 

1 006 

92 

33 -3 

O.924 

3 

—16 .1 

I . IOI 

48 

8 .9 

I .004 

93 

33 -9 

0.923 

4 

— 15 .6 

1.099 

49 

9 -4 

1.002 

94 

34 -4 

0.921 

5 

-15 .0 

1.096 

5 ° 

10 .0 

1.000 

95 

35 -° 

0.919 

6 

-14 .4 

1•° 94 | 

51 

10 .6 

0.998 

96 

35 6 

0.917 

7 

-13 -9 

1.092 

52 

11 .1 

0.996 

97 

36 .1 

0.916 

8 

-13 -3 

1.089 

53 

11 .7 

0.994 

98 

36 -7 

0.914 

9 

—12 .8 

I .O87] 

54 

12 .2 

0.992 

99 

37 2 

0.912 

IO 

— 12 .2 

1.085 

55 

12 .8 

0.990 

IOO 

37 -8 

O.9I I 

11 

-11 .7 

1.082 

56 

i 3 -3 

0.988 

IOI 

3 s -3 

O.9O9 

12 

— 11 .1 

1.080 

57 

13 -9 

0.986 

102 

^8 .y 

0.908 

!3 

— 10 .6 

1.078 

58 

14 -4 

0.985 

103 

39 -4 

0.906 

14 

—10 .0 

1.076 

59 

15 -o 

0.983 

104 

40 .O 

0.905 

15 

— 9 .4 

1 °73 

60 

15 -6 

0.981 

105 

40 .6 

0.903 

16 

— 8 .9 

1.071 

6l 

16 .1 

0.979 

106 

41 .1 

0.902 

17 

- 8 .3 

1 .069 

62 

16 .7 

0.977 

IO7 

4 i -7 

O.9OO 

18 

— 7 .8 

1.067] 

63 

17 .2 

0.975 

108 

42 .2 

0.899 

*9 

- 7 .2 

1.064 

64 

17 -8 

o -973 

IOg 

42 .8 

0.897 


Temp. 

Fahr. 

Temp. 

Cent. 

C D 

IIO° 

43°-3 

0.895 

hi 

43 -9 

0.894 

112 

44 -4 

0.892 

”3 

45 -o 

0.891 

114 

45 -6 

0.890 

115 

46 .1 

0.888 

Si 6 

46 .7 

0.886 

”7 

47 -2 

0.885 

118 

47 -8 

0.884 

119 

48 .3 

0.882 

120 

48 .9 

0.881 

121 

49 -4 

0.880 

122 

50 .0 

0.878 

123 

50 .6 

0.877 

124 

5 i -i 

0.876 

125 

5 i -7 

0.874 

126 

52 .2 

0.873 

127 

52 .8 

0.871 

128 

53 -3 

0.870 

129 

53 -9 

0.868 

13° 

54 -4 

0.867 


Table LIII. 


Correction (Ca) to 
Mean Refraction 
Depending upon 
Reading of At¬ 
tached Thermom¬ 
eter. 


A = bKC£>)(C a ). 

(From Hayford’s 
Geodetic Astronomy.) 


Temp. 

Fahr. 

Temp. 

Cent. 

CA 

—3°° 

-34°.4 

I .007 

—20 

-28 .9 

1.006 

— IO 

-23 -3 

1 -005 

0 

— 17 .8 

1.005 

+10 

— 12 .2 

I .004 

20 

- 6 .7 

1.003 

3 ° 

— 1 .1 

1.002 

40 

+ 4-4 

I .OOI 

50 

IO .O 

I .OOO 

60 

15 -6 

0.999 

70 

21 . I 

0.998 

80 

26 .7 

0.997 

90 

32 .2 

0 996 

100 

37 -8 

0.996 

1 IO 

43 -3 

0.995 

120 

48 .9 

0.994 

130 

54 -4 

0.993 

_ 





















































CHAPTER XXXII. 


TIME. 

303. Interconversion of Time —Sidereal time is referred 
to a fixed star, mean time to the sun. There is one more 
sidereal than solar day in the year. 

366.24 sidereal days = 365.24 mean solar days. 

24 hrs. sidereal time = 23 hrs. 56 min. 04.091 sec. mean 
solar time. 

24 hrs. mean time = 24 hrs. 03 min. 56.555 sec. sidereal 
time. 


Relations of Sidereal , Civil , and Mean Solar Time. 


T s — a s — I when T s > .(4 16) 

24 h + T s — 01,-1 when T s < a,; . . . . (117) 

T s = <* s + T m ;.(u8) 

T m = /— C s for D t \ .(119) 

T m = T for D t if T m < I2 h ;.(120) 

T* = T s — ^.(121) 

T~ T m - I2 h for A + / if 7 ^ > I2 h . . . (122) 

For example see Article 315. 

Relation of Sidereal Time to Right Ascension and 
Hour-angle of a Star. 

T s = (x + t . .. (123) 

and t — T s — a .( I2 4) 

Relations of Sidereal and Mean Solar Intervals of Time. 


—Let 1 = ratio of tropical year, expressed in sidereal day to 
tropical year expressed in mean solar day; then 


695 









6g6 


TIME, 


Table LIV. 

CONVERSION OF MEAN TIME INTO SIDEREAL TIME. 

(From Smithsonian Geographical Tables.) 


s 

m 

0 

m 

1 

m 

2 

m 

3 


h 

m 

s 

h 

m 

s 

h 

m 

s 

h 

in 

s 

0 

O 

O 

0 

6 

5 

*5 

12 

10 

29 

18 

15 

44 

I 

O 

6 

5 

6 

I I 

20 

12 

16 

34 

18 

21 

49 ' 

2 

O 

12 

IO 

6 

17 

25 

12 

22 

40 

18 

27 

54 

3 

O 

18 

l 6 

6 

23 

3 ° 

12 

28 

45 

18 

33 

59 

4 

O 

24 

21 

6 

29 36 

12 

34 

50 

18 

40 

5 

5 

O 

30 

26 

6 

35 

41 

T 2 

40 

55 

18 46 

10 

6 

O 

36 

3 1 

6 

41 

36 

12 

47 

1 

18 

52 

45 

7 

O 

42 

37 

6 

47 

5 » 

12 

53 

6 

18 58 

20 

8 

O 

48 

42 

6 53 56 

12 

59 

I I 

49 

4 

26 

9 

O 

54 

47 

6 

O 

2 

43 

5 

l 6 

19 

IO 

21 

70 

I 

O 

52 

7 

6 

7 

13 

1 I 

21 

19 

16 36 

I I 

I 

6 58 

7 

1 2 

12 

13 

17 

27 

49 

22 

44 

12 

I 

13 

0 

7 

18 

17 

13 

23 

32 

>9 

28 

47 

13 

I 

19 

8 

7 

24 

23 

43 

29 

37 

19 

34 

52 

14 

I 

25 

*3 

7 

30 

28 

43 

35 

42 

19 

40 

57 

15 

1 

3 1 

19 

7 36 33 

13 

41 

48 

>9 

47 

2 

l 6 

I 

37 

24 

7 

42 

38 

13 

47 

53 

19 

53 

7 

17 

I 

43 

29 

7 48 

44 

13 

53 58 

19 

59 

13 

18 

I 

49 

34 

7 

54 

49 

14 

O 

3 

20 

5 

t 8 

19 

I 

55 

40 

8 

O 

54 

14 

6 

9 

20 

I I 

23 

20 

2 

1 

45 

8 

6 

59 

14 

12 

>4 

20 

1 7 

28 

21 

2 

7 

50 

8 

*3 

5 

'4 

18 

19 

20 

23 

34 

22 

2 

13 

55 

8 

19 

IO 

14 

24 

24 

20 

29 

39 

23 

2 

20 

I 

8 

25 

15 

44 

?° 

30 

20 

35 

44 

24 

2 

26 

6 

8 

3 > 

20 

14 3 6 

35 

20 

41 

49 

25 

2 

32 

11 

8 

37 

26 

44 

42 

40 

20 

47 

55 

26 

2 

38 16 

8 

43 

31 

44 

48 

45 

20 

54 

O 

27 

2 

44 

22 

8 

49 36 

14 

54 

54 

21 

O 

5 

28 

2 

50 

27 

8 

55 

41 

45 

O 

56 

21 

6 

IO 

29 

2 

56 

32 

9 

I 

47 

45 

7 

I 

21 

12 

16 

30 

3 

2 

37 

9 

7 

52 

45 

43 

6 

21 

18 

21 

31 

3 

8 

43 

9 

r 3 

57 

45 

49 

12 

21 

24 

26 1 

32 

3 

14 48 

9 

20 

2 

•5 

25 

17 

21 

30 

31 

33 

3 

20 

53 

9 

26 

8 

45 

3 ' 

22 

21 

36 37 

34 

3 

26 58 

9 

32 

13 

45 

37 

27 

21 

42 

42 

35 

3 

33 

3 

9 38 

18 

45 

43 

33 

21 

48 47 

3 6 

3 

39 

9 

9 

44 

23 

45 

49 38 

21 

54 

52 

37 

* 3 

45 

14 

9 

5 « 

28 

15 

55 

43 

22 

O 

= 8 

38 

3 

51 

19 

9 56 

34 

l 6 

I 

48 

22 

7 

3 

39 

3 

57 

?4 

IO 

2 

39 

l 6 

7 

54 

22 

13 

8 

40 

4 

3 

30 

IO 

8 

44 

l 6 

43 

59 

22 

19 

13 

4 i 

4 

9 

35 

IO 

•4 

49 

l 6 

20 

4 

22 

25 

19 

42 

4 

15 

40 

IO 

20 

55 

l 6 

26 

9 

22 

34 

24 

43 

4 

21 

45 

IO 

27 

O 

l 6 

32 

14 

22 

37 

29 

44 

4 

27 

5 i 

IO 

33 

5 

j6 38 

20 

22 

43 

34 

45 

4 

33 50 

IO 

39 

IO 

l 6 

44 

25 

22 

49 

39 

46 

4 

40 

I 

IO 

45 

16 

l 6 

50 

30 

22 

55 

45 

47 

4 

46 

6 

IO 

51 

21 

16 56 

35 

2 3 

I 

5 ° 

48 

4 

52 

12 

IO 

57 

26 

17 

2 

41 

23 

7 

55 

49 

4 

58 

17 

11 

3 

3 1 

17 

8 46 

23 

14 

O 

5 ® 

5 

4 

22 

11 

0 

37 

47 

44 

51 

23 

20 

6 

5 l 

5 

10 

27 

11 

*5 

42 

47 

20 

56 

23 

26 

n ! 

52 

5 

16 

33 

II 

21 

47 

17 

27 

2 

23 

32 

l 6 

53 

5 

22 

38 

11 

27 

52 

>7 

33 

7 

23 38 

21 

51 

5 

28 

43 

II 

33 

58 

17 

39 

12 

23 

44 

27 

55 

5 

34 4 » 

11 

40 

3 

17 

45 

47 

23 

50 

32 

56 

5 

40 

54 

11 

46 

8 

47 

51 

23 

23 56 

37 

57 

5 

46 

59 

11 

52 

13 

17 

57 

28 

24 

2 

42 

58 

5 

53 

4 

11 

58 

>9 

18 

3 

33 

24 

8 48 

59 

5 

59 

Q 

12 

4 

24 

18 

9 38 

24 

44 

S 3 

60 

6 

5 

*5 I 

12 

IO 

29 

18 

15 

44 

24 

20 

58 


s 

m s 

s 

m s 

0.00 

O O 

0.50 

3 3 

0 

0 

M 

0 4 

0.51 

3 6 

0.02 

0 7 

0.52 

3 10 

0.03 

Oil 

0.53 

3 14 

O.O4 

0 15 

0.54 

3 17 

0.05 

0 18 

055 

3 21 

0 06 

O 22 

0.56 

3 25 

O.O7 

0 26 

o-57 

3 28 

0.08 

0 29 

0.58 

3 32 

O.09 

0 33 

0 59 

3 35 

O. TO 

0 37 

0.60 

3 39 

O. I 1 

O 40 

0.61 

3 43 

O. 12 

0 44 

0.62 

3 46 

0.13 

0 47 

0.63 

3 50 

O.I4 

0 51 

0.64 

3 54 

0 15 

0 55 

0.65 

3 57 

0.16 

0 58 

0.66 

4 1 

0.17 

I 2 

0.67 

4 5 

0.18 

i 6 

0 68 

4 8 

0 19 

« 9 

0.69 

4 12 

O 20 

113 

0 

t"-. 

d 

4 16 

0.21 

117 

0.71 

4 19 

0.22 

1 20 

0.72 

4 23 

0.23 

124 

0.73 

4 27 

O 24 

1 28 

0.74 

4 30 

0.25 

1 3 1 

o-75 

4 34 

0.26 

1 35 

0.76 

4 38 

0.27 

1 39 

0.77 

4 4 1 

0.28 

I 42 

0.78 

4 45 

0.29 

1 46 

0.79 

4 49 • 

0.30 

1 so 

0 

00 

0 

4 52 

O.3I 

1 53 

0.81 

4 56 

0.32 

1 57 

0.82 

4 59 

o-33 

2 I 

0.83 

5 3 

o-34 

2 4 

0.84 

5 7 

0.35 

2 8 

0.85 

5 >c* 

0.36 

2 II 

0.86 

5 14 

o-37 

2 15 

0.87 

5 18 

0.38 

2 19 

0.88 

5 21 

o-39 

2 22 

0 89 

5 25 

0.40 

2 26 

0.90 

5 29 

0.41 

2 30 

O.9I 

5 32 

O. 42 

2 33 

0.92 

5 3 6 

o-43 

2 37 

°-93 

5 40 

0.44 

2 41 

0.94 

5 43 

0 45 

2 44 

o.95 

5 47 

0.46 

2 48 

o.q6 

5 5i 

0 47 

2 52 

0.97 

5 54 

0 48 

2 55 

0.98 

5 58 

0.49 

2 58 

0.99 

6 2 

0 50 

3 3 

I OO 

6 5 


be i4 h 57 m 


- Y» s / 3 28 -56. 

The table gives 

first for i4 h 54 ra 
then for 2 41 


mean time 


m 


278 

0.44 


The sum 
1 4 h 57 m 32*.56 + 2 m 27^.44 = 
is the required sidereal time. 


2 27.44 


i5 h o D 














































































































































INTERCON VERSION OF TIME 


697 


Table LV. 

CONVERSION OF SIDEREAL TIME INTO MEAN TIME. 



m 

m 

in 

s 

0 

I 

2 


h m s 

h m s 

h in s 

0 

000 

6 6 15 

12 12 29 

I 

066 

6 12 21 

12 18 35 

2 

O 12 12 

61827 

12 24 42 

3 

0 18 19 

6 24 33 

12 30 48 

4 

0 24 25 

6 30 40 

12 36 54 

5 

0 30 31 

6 36 46 

12 43 0 

6 

0 36 37 

6 42 52 

12 49 7 

7 

O 42 44 

6 48 58 

12 55 13 

8 

0 48 50 

6 55 4 

13 1 19 

9 

0 54 56 

7 111 

'3 7 25 

TO 

I I 2 

7717 

! i 3 13 31 

I I 

x 7 9 

713 23 

13 >9 38 

12 

1 13 15 

7 19 29 

13 25 44 

»3 

I 19 21 

7 25 36 

13 3 i 5 ° 

14 

1 23 27 

7 3142 

13 37 56 

15 

1 31 34 

7 37 48 

13 44 3 

16 

1 37 40 

7 43 54 

13 5 ° 9 

17 

x 43 46 

7 50 1 

13 56 15 

l8 

1 49 52 

7 56 7 

14 2 21 

19 

1 55 59 

8213 

14 8 28 

20 

225 

O' 

00 

00 

14 14 34 

21 

2 811 

8 14 26 

14 20 40 

22 

2 14 17 

8 20 32 

14 26 46 

23 

2 20 24 

8 26 38 

14 3 2 53 

24 

2 26 30 

8 32 44 

i 4 38 59 

25 

2 32 36 

8 38 51 

14 45 5 

26 

2 38 42 

8 44 57 

14 51 n 

27 

2 44 49 

8 51 3 

14 57 18 

28 

2 50 55 

8 57 9 

15 3 24 

29 

2 57 1 

9 3 16 

15 9 30 

30 

337 

9 9 22 

15 15 36 

31 

3 9 14 

9 15 28 

15 21 43 

32 

3 *5 20 

9 21 34 

15 27 49 

33 

3 21 26 

9 27 41 

15 33 55 

34 

3 27 32 

9 33 47 

15 40 i 

35 

3 33 38 

9 39 53 

15 46 8 

36 

3 39 45 

9 45 59 

15 5 2 M 

37 

3 45 5 i 

9 52 5 

15 58 20 

38 

3 5 i 57 

9 58 12 

16 4 26 

39 

3 58 3 

10 4 18 

16 -10 33 

40 

4 4 10 

10 10 24 

16 16 39 

41 

4 10 16 

10 16 30 

16 22 45 

42 

4 16 22 

IO 22 37 

16 2S 51 

43 

4 22 28 

10 28 43 

16 34 57 

44 

4 28 35 

10 34 49 

16 41 4 

45 

4 34 4 i 

10 40 55 

16 47 10 

46 

4 40 47 

TO 47 2 

16 53 16 

47 

4 46 53 

10 53 8 

16 59 22 

48 

4 53 0 

10 59 14 

17 5 2 9 

49 

4 59 6 

IT 5 20 

17 11 35 

50 

5 5 12 

II IT 27 1 

17 17 41 

5 1 

5 11 18 

i 1 * 7 33 j 

17 2 3 47 

52 

5 17 25 

11 23 39 

17 29 54 

53 

5 23 31 

11 29 45 

17 36 0 

54 

5 29 37 

if 35 52 

17 42 6 

55 

5 35 43 

11 41 58 

17 48 12 

56 

5 4 i 5 ° 

11 48 4 

17 54 19 

57 

5 47 56 

it 54 10 

18 0 25 

58 

5 54 2 

12 O 17 

18 6 31 

=9 

608 

12 623 

18 12 37 

60 

66x5 

12 12 29 • 

18 18 44 


m 

3 





h m s 

s 

m s 

s 

m s 

18 18 44 

0.00 

O O 

0.50 

3 3 

18 24 50 

0.01 

0 4 

0.51 

3 7 

18 30 56 

0.02 

0 7 

0.52 

3 i° 

1837 2 

0.03 

O II 

o-53 

3 14 

18 43 9 

O 04 

0 15 

°-54 

3 18 

18 49 15 

0.05 

0 18 

0 55 

3 21 

18 £5 21 

0.06 

O 22 

0.56 

3 25 

19 1 27 

0.07 

0 26 

0.57 

3 29 

i9 7 34 

1 0.08 

O 29 

0.58 

3 32 

i9 13 40 

o.cg 

0 33 

0 59 

3 36 

19 19 49 

0.10 

1 0 37 

0.60 

3 40 

19 25 52 

0.11 

O 40 

0.61 

3 43 

J 9 3i 59 

012 

0 44 

0.62 

3 47 

>9 38 5 

0.13 

0 48 

0 63 

3 5i 

19 44 11 

0.14 

0 51 

0.64 

3 54 

19 50 17 

0.15 

0 55 

0.65 

3 58 

19 56 23 

0.16 

0 59 

0.66 

4 2 

20 2 30 

0.17 

I 2 

0. 67 

4 5 

20 8 36 

0. i§ 

i 6 

0.68 

4 9 

20 14 42 

O. IQ 

t IO 

0.60 

4 13 

20 20 48 

0.20 

1 T 3 

0.70 

4 16 

20 26 55 

0.21 

117 

0.71 

4 20 

20 33 1 

0.22 

1 21 

O. 72 

4 24 

20 39 7 

0.23 

124 

o-73 

4 27 

20 45 13 

O.24 

128 

0.74 

4 31 

20 51 20 

0.25 

132 

o-75 

4 35 

20 57 26 

0.26 

135 

0.76 

4 38 

21 3 32 

O.27 

139 

0.77 

4 42 

21 9 38 

0.28 

143 

0.78 

4 46 

21 15 45 

O.29 

146 

O.7Q 

4 49 

2T 21 51 

O . ^O 

150 

0 80 

4 53 

21 27 57 

0.31 

154 

0.81 

4 57 

21 34 3 

0.32 

1 57 

0.82 

5 0 

21 40 IO 

0.33 

2 1 

0.83 

5 4 

21 46 16 

o-34 

2 5 

0.84 

5 8 

21 52 22 

c.35 

2 8 

0.85 

5 11 

21 58 28 

0.36 

2 12 

0.86 

5 15 

22 4 35 

°’ 3 l 

2 16 

0.87 

5 19 

22 TO 41 

0 38 

2 19 

0.88 

5 22 

22 16 47 

0.39 

2 23 

0.89 

5 26 

22 22 53 

0.40 1 

2 26 1 

O.9O 

5 30 

22 29 O 

| °‘4 l 

2 30 

O.9I 

5 33 

22 35 6 

0.42 

2 34 

0.92 

5 37 

22 41 12 

0-43 

2 37 

o.93 

5 4i 

22 47 18 

0 44 

2 41 

0.94 

5 44 

22 53 24 

o.45 

2 45 

0-95 

5 48 

22 59 31 

0 46 

2 48 

0.96 

5 52 

23 5 37 

°-47 

2 52 

0.97 

5 55 

23 11 43 

0.48 

2 56 

0.98 

5 59 

23 17 49 

0.49 

2 59 1 

0.99 

6 3 

23 23 56 

0.50 

3 3 

I OO 

6 6 

23 30 2 

23 3 6 8 

23 42 I4 

23 48 21 

23 54 27 

Example . — Given 
The table gives 

first for i4 h 57 ra 

[5* 1 o m o 8 . 

18 8 2® 

27® 

then for 2 

42 

0.44 

24 0 33 

24 6 39 

24 12 46 

24 18 52 

15 0 0 2 27 44 

The difference 

15* 1 o m o s — 2 m 27 s .44 = i4 h 57 m 32®.s6 

24 24 58 

is the required mean 

time 










































































































































































698 


TIME. 


366.2422 

365.2422 


1.002738; 


/'= rl — /+ (r — i)/ = 1 + 0.002738/; . . 

I = r~ x I’ — I — (1 — r" 1 )/' — I' — 0.002730/'. 


(125) 

(126) 



Fig. 180.—Conversion of Time. 

These conversions are made by means of Tables LIV and 
LV, in which examples are shown. 

304. Interconversion of Time and Arc. —In the perform¬ 
ance of geodetic operations it frequently becomes necessary 
to convert time into longitude expressed in degrees of arc, 
and vice versa. The following tables facilitate this and sim¬ 
ilar operations and are for the interconversion of sidereal time 
and parts of the equator in degrees of arc, or of sidereal time 
and terrestrial longitude in arc. 


Table LVI. 

CONSTANTS FOR THE INTERCONVERSION OF TIME AND ARC. 


r 

-- 

Logarithms. 

12 hours, expressed in seconds. = 43200. 

Complement to the same. = .00002315 

24 hours, expressed in seconds. = 86400. 

Complement to the same. = .00001157 

360 degrees, expressed in seconds. = 1296000. 

To convert sidere>al time into mean solar time.. 

4-6354837 

5.3645163 

4-9365137 

5.0634863 

6.1126050 

9.9988126 















IN TER CON VERSION OF TIME AND ARC . 699 

Table LVII. 

CONVERSION OF TIME INTO ARC OR TERRESTRIAL 

LONGITUDE. 


(From Lee’s Tables.) 


Hours. 

Minutes. 


Seconds. 


Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

h. 

O 

in. 

c / 

m. 

0 / 

s. 

t // 

s. 

/ // 

I 

15 

I 

O 15 

31 

7 45 

I 

O 15 

31 

7 45 

2 

30 

2 

O 30 

32 

8 00 

2 

O 30 

32 

8 00 

3 

45 

3 

0 45 

33 

8 15 

3 

0 45 

33 

8 15 

4 

60 

4 

1 00 

34 

8 30 

4 

1 00 

34 

8 30 

5 

75 

5 

1 15 

35 

8 45 

5 

1 15 

35 

8 45 

6 

90 

6 

1 3 ° 

36 

9 00 

6 

1 30 

36 

9 00 

7 

105 

7 

1 45 

37 

9 15 

7 

1 46 

37 

9 15 

8 

120 

8 

2 00 

38 

9 30 

8 

2 00 

38 

9 30 

9 

135 

9 

2 15 

39 

9 45 

9 

2 15 

39 

9 45 

10 

150 

10 

2 30 

40 

10 00 

10 

2 30 

40 

10 00 

11 

165 

11 

2 45 

4 i 

10 15 

II 

2 45 

4 i 

10 15 

12 

180 

12 

3 00 

42 

10 30 

12 

3 00 

42 

10 30 

13 

195 

13 

3 15 

43 

10 45 

13 

3 15 

43 

10 4* 

14 

210 

14 

3 30 

44 

11 00 1 

14 

3 30 

44 

11 00 

15 

225 

15 

3 45 

45 

11 15 

15 

3 45 

45 

11 15 

16 

240 

16 

4 00 

46 

11 30 

16 

4 00 

46 

11 30 

17 

255 

17 

4 15 

47 

11 45 

17 

4 15 

' 47 

11 45 

18 

270 

18 

4 30 

48 

12 00 

18 

4 30 

48 

12 00 

19 

285 

19 

4 45 

49 

12 15 

19 

4 45 

49 

12 15 

20 

300 

20 

5 00 

50 

12 30 

20 

5 00 

50 

12 30 

21 

315 

21 

5 15 

5 i 

12 45 

21 

5 15 

5 i 

12 45 

22 

330 

22 

5 30 

52 

13 00 

22 

5 30 

52 

13 00 

23 

345 

23 

5 45 

53 

13 15 

23 

5 45 

53 

13 15 

24 

360 

24 

6 00 

54 

13 30 

24 

6 00 

54 

13 30 

25 

6 15 

55 

13 45 

25 

6 15 

55 

13 45 



26 

6 30 

56 

14 00 

26 

6 30 

56 

14 00 



27 

6 45 

57 

14 15 

27 

6 45 

57 

14 15 



28 

7 00 

58 

14 30 

28 

7 00 

58 

14 30 



29 

7 15 

59 

14 45 

29 

7 15 

59 

14 45 

L 


30 

7 30 

60 

15 00 

30 

7 30 

60 

15 00 

_ 















































































700 


TIME. 


Table LVIII. 

CONVERSION OF TIME INTO ARC, ETC.— {Continued.) 

(From Lee’s Tables.) 


Tenths of Seconds. 

J2 c« 

•a § 4 i 

5 8 e 

Arc. 
























Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

0 ^ V*-, 

^00 

H 


s. 

// 

s. 

// 

s. 

// 

I 

s. 

// 

s. 

// 

s. 

// 

0.01 

0.15 

0.21 

3-15 

O.41 

6.15 

0.61 

9 -i 5 

O.81 

12.15 

0.001 

0.015 

0.02 

0.30 

0.22 

3-30 

O.42 

6.30 

0.62 

9-30 j 

0.82 

12.30 

0.002 

O.030 

0.03 

0-45 

O.23 

3-45 

0-43 

6.45 

0.63 

9-45 

0.83 

12.45 

0.003 

O.045 

O.04 

O.60 

O.24 

3.60 

O.44 

6.60 

0.64 

9.60 

0.S4 

12 60 

O.OO4 

0.060 

0.05 

0-75 

0.25 

3-75 

0.45 

6.75 

0.65 

9-75 

085 

T2.75 

0.005 

O.075 

O.06 

0.90 

0.26 

3-90 

O.46 

6 90 

0.66 

9.90 

0.86 

12. go j 

0.C06 

o.cgo 

0.07 

I.05 

0.27 

4-05 

0-47 

7-05 

0.67 

10.05 | 

0.87 

13-05 

0.007 

0.105 

O.08 

1.20 

0.28 

4.20 

O.48 

7.20 

0.68 

10.20 

0.88 

13.20 

O.OOS 

0.120 

O.09 

i -35 

O.29 

4-35 

O.49 

7-35 

0.69 

10-35 

c.89 

13-35 

O.OO9 

o.i 35 

O. TO 

1.50 

O.30 

4-50 

O.50 

7-50 

0.70 

10.50 

0.90 

13-50 

0.010 

0.150 

O II 

1.65 

O.31 

4-65 

0.51 

7-65 

0.71 

10.65 j 

0.91 

13-65 



0.12 

1.80 

O.32 

4.80 

O.52 

7.80 

0.72 

10 80 

0.92 

13.80 



0.13 

i -95 

0-33 

4-95 

0-53 

7-95 

0-73 

10.95 

o -93 

13-95 



0.14 

2.10 

0.34 

5.10 

0-54 

8.10 

0.74 

11.10 

0.94 

14.10 



0.15 

2.25 

0.35 

5-25 

0 55 

8.25 

0-75 

H -25 

1 

0-95 

14 25 



0.16 

2.40 

O.36 

5-40 

0.56 

8.40 

0.76 

11.40 

0.96 

14.40 



0.17 

2-55 

0-37 

5-55 

0-57 

8-55 

0.77 

n -55 

0.97 

14-55 



O. l8 

2.70 

O.38 

5-70 

0.58 

8.70 

0.78 

11.70 } 

0.98 

14.70 



O. ig 

2.85 

0-39 

5-85 

0.59 

8.85 

0.79 

11.85 

0.99 

14.85 



0.20 

3.00 

O.40 

6.00 

1 

0.60 

9.00 

0.80 

12.00 

1.00 

15-00 




305* Determination of Time.—This consists in finding 
the correction to the dock, watch, or chronometer used to 
record time. 

Let T — clock time of any event at any place; 

AT — clock correction ; and 
T 0 = required corresponding true time. Then 


T 0 — T + AIT. .(127) 

Clock correction may be found by several methods, three 
of the best and simplest being those given by Prof. R. S. 


































































DETERM1NA TION OF TIME 


701 


Table LIX.—conversion of arc into time. 


(From Smithsonian Geographical Tables.) 


0 

h. m. 

0 

h. m. 

O 

h. m. 

O 

h. m. 

O 

h. m 

O 

h. m 

t 

m. s. 

// 

s. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 0 

0 4 
0 8 
0 12 
0 16 
0 20 
0 24 
0 28 
0 32 
0 36 

6 C 

61 

6; 

6: 

64 

6 a 

66 

67 

6£ 

6q 

1 4 0 

4 4 

4 8 
4 12 
4 16 
4 20 
4 24 
4 28 
4 32 
4 36 

12 ( 

12] 

122 

12; 

124 

125 

i2(: 

I2 7 

I2g 

125 

> 8 

8 

8 

i 8 is 

8 i< 
8 2 C 

8 2 
8 2i 
8 3 ‘ 

8 36 

d 184 

4 18 
3 18' 

18 

5 i8< 
> 18 -' 

i8( 

'• 18; 
188 
18c 

► 12 

12 

12 1 
12 1 J 
l 12 l( 
12 2 C 

12 24 
12 28 
12 35 
12 3 6 

3 21 

4 24 
1 24 

24 

5 24 
21 . 

24 

24 

24I 

24c 

25 ( 

) 16 

1 16 

2 16 

16 1 
16 n 

> l6 2 C 

5 16 2 

1 16 2I 

16 

16 3 < 

o 30 

4 30 
3 30 
\ 30 

5 30 
3 30 , 
1 30 

30 

30! 

30c 

) 20 c 

1 20 4 

2 20 f 

3 20 12 

4 20 16 

5 20 20 
5 20 24 
7 20 2? 
3 20 32 
) 20 36 

0 

1 

I 2 

3 

4 

5 

6 

7 

8 

9 

10 

0 0 

0 4 

0 8 
0 12 
0 16 
0 20 

0 24 
0 28 
0 32 
0 36 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.000 

0.067 

°- x 33 ( 
0.200 
O.267 

°* 333 
0.400 
0.467 
o -533 
0.600 

10 

O 40 

70 

4 40 

130 

8 4c 

190 

12 4c 

• 16 4c 

>310 20 40 

O 40 

1 10 

0.667 

11 

12 

x 3 

M 

15 

16 

*7 

18 

0 44 
0 48 
0 52 
0 56 

1 0 

I 4 

1 8 

1 12 

1 16 

71 

72 

73 

74 

75 

76 

77 

78 

79 

4 44 
4 48 
4 52 

4 5 6 

5 0 

5 4 

5 8 

5 12 

5 16 

1 3 1 

132 

| x 33 
1 84 

135 

136 
*37 
138 

x 39 

8 44 
8 48 
8 52 

8 56 

9 0 
9 4 
9 8 
9 12 
9 16 

191 

192 

193 

194 

195 

196 

197 

198 
x 99 

12 44 
12 48 
12 52 

12 56 
x 3 0 
x 3 4 
x 3 8 
x 3 X 2 

13 16 

253 
25= 
25: 

254 
25 ( 

256 

257 

255 
259 

16 4; 
16 48 
16 52 

16 56 

17 0 
17 4 
17 8 
17 12 
17 16 

311 

312 

3 X 3 

3 X 4 

31 a 

316 

3 1 7 

3 1 8 
3 X 9 

20 44 
; 20 48 
20 52 

20 56 

21 0 

21 4 

21 8 

21 12 
21 16 

11 

12 

x 3 

T 4 

15 

16 

1 7 

18 

x 9 

0 44 
0 48 
0 52 

0 56 

1 0 

x 4 

1 8 

1 12 

1 16 

11 

12 

1 13 

x 4 

15 

16 

x 7 

18 

x 9 

°-733 

0.800 

0.867 

0 933 

1.000 

1.067 

1 • x 33 

1.200 
1.267 

20 

1 20 

SO 

5 20 

110 

9 20 

200 

13 20 

260 

17 20 

320 

21 20 

20 

I 20 

20 

x -333 

21 

I 24 

81 

5 24 

141 

9 24 

201 

1 3 24 

261 

17 24 

321 

21 24 

21 

1 24 

21 

.I.400 

22 

I 28 

82 

5 28 

142 

9 28 

202 

13 28 

262 

17 28 

322 

21 28 

22 

i 28 

22 

1.467 

2 3 

1 32 

83 

5 32 

x 43 

9 32 

203 

13 32 

263 

17 32 

323 

21 32 

23 

1 32 

23 

1 • 533 

24 

1 36 

84 

5 36 

x 44 

9 36 

204 

x 3 36 

264 

x 7 36 

324 

1 21 36 

24 

1 36 

24 

1.600 

25 

I 40 

85 

5 40 

145 

9 40 

205 

x 3 40 

265 

17 40 

325 

21 40 

25 

I 40 

25 

1.667 

26 

1 44 

86 

5 44 

1:46 

9 44 

206 

x 3 44 

266 

x 7 44 

326 

21 44 

26 

i 44 

26 

1 733 

27 

1 48 

87 

5 48 

x 47 

9 48 

207 

x 3 48 

267 

17 48 

327 

21 48 

27 

1 48 

27 

1.800 

28 

1 52 

88 

5 52 

148 

9 52 

208 

x 3 52 

268 

x 7 52 

328 

21 52 

28 

x 52 

28 

1.867 

29 

x 56 

89 

5 56 

x 49 

9 56 

209 

x 3 56 

269 

x 7 56 

329 

21 56 

29 

x 56 

29 

1 -933 

30 

2 O 

90 

6 0 

150 

10 0 

210 

14 O 

270 

18 0 

330 

22 O 

30 

2 O 

30 

2.000 

31 

2 4 

91 

6 4 

x 5 1 

10 4 

2 11 

1 4 4 

271 

18 4 

33 1 

22 4 

3 X 

2 4 

3 X 

2.067 

32 

2 8 

92 

6 8 

x 5 2 

10 8 

2 T 2 

14 8 

272 

18 8 

332 

22 8 

32 

2 8 

32 

2 - x 33 

33 

2 12 

93 

6 12 

x 53 

IO 12 

213 

14 12 

273 

18 12 

333 

22 12 

33 

2 12 

33 

2.200 

34 

2 16 

94 

6 16 

x 54 

10 16 

214 

14 16 

274 

18 16 

334 

22 16 

34 

2 16 

34 

2.267 

35 

2 20 

95 

6 20 

155 

IO 20 

215 

14 20 

275 

18 20 

335 

22 20 

35 

2 20 

35 

2 -333 

36 

2 24 

96 

6 24 

156 

10 24 

216 

14 24 

276 

18 24 

336 

22 24 

36 

2 24 

36 

2.400 

37 

2 28 

97 

6 28 

x 57 

IO 28 

2 T 7 

14 28 

277 

18 28 

337 

22 28 

37 

2 28 

37 

2.467 

38 

2 32 

98 

6 32 

x 58 

IO 32 

218 

x 4 32 

278 

18 32 

338 

22 32 

38 

2 32 

38 

2-533 

39 

2 36 

99 

6 36 

x 59 

IO 36 

219 

x 4 36 

279 

18 36 

339 

22 36 

39 

2 36 

39 

2.600 

40 

2 40 

100 

6 40 

160 

IO 40 

220 

14 40 

280 

18 40 

310 

22 40 

10 

2 40 

10 

2.667 

41 

2 44 

IOl 

6 44 

l6l 

IO 44 

221 

x 4 44 

28l 

18 44 

34 x 

22 44 

4 X 

2 44 

4 X 

2-733 

42 

2 48 

102 

6 48 

162 

10 48 

222 

x 4 48 

282 

18 48 

342 

22 48 

42 

2 48 

42 

2.800 

43 

2 52 

103 

6 52 

163 

IO 52 

223 

x 4 52 

283 

18 52 

343 

22 52 

43 

2 52 

43 

2.867 

44 

2 56 

IO4 

6 56 

164 

IO 56 

224 

x 4 56 

284 

18 56 

344 

22 56 

44 

2 56 

44 

2-933 

45 

3 0 

105 

7 0 

165 

II 0 

225 

x 5 0 

285 

19 0 

345 

23 O 

15 

3 0 

15 

3.000 

46 

3 4 

106 

7 4 

166 

II 4 

226 

x 5 4 

286 

19 4 

346 

23 4 

46 

3 4 

46 

3.067 

47 

3 8 

IO7 

7 8 

167 

11 8 

227 

x 5 8 

287 

19 8 

34 7 J 

23 8 

47 

3 8 

47 

3 • x 33 

48 

3 12 

108 

7 12 

168 

II 12 

228 

15 12 

288 

19 T 2 

348 

23 12 

48 

3 12 

48 

3.200 

49 

3 16 

IO9 

716 

169 

11 16 

229 

15 16 

289 

19 16 

349 : 

23 16 

49 

3 16 

49 

3.267 

50 

3 20 

110 

7 20 

170 

11 20 

230 

15 20 

290 

19 20 

350 

23 20 

50 

3 20 

50 

3-333 

51 

3 24 

III 

7 24 

X 7 X 

II 24 

231 

x 5 24 

29I 

19 24 

35 x 

23 24 

5 X 

3 24 

5 X 

3.400 

52 

3 28 

I 12 

7 28 

172 

11 28 

232 

15 28 

292 

19 28 

352 

23 28 

52 

3 28 

52 

3-467 

53 

3 32 

xx 3 

7 32 

x 73 

li 32 

233 

x 5 32 

293 

IQ 32 

353 

23 32 

53 

3 32 

53 

3-533 

54 

3 36 

114 

7 36 

x 74 

11 36 

234 

x 5 3 6 

294 

I 9 36 

354 

23 3 6 

54 

3 3 6 

54 

3.600 

00 

3 40 

1 15 

7 40 

175 

11 40 

235 

x 5 40 

295 

x 9 40 

355 

23 40 

55 

3 40 

55 

3.667 

56 

3 44 

Il6 

7 44 

176 

ii 44 

236 

x 5 44 

296 

19 44 

356 

23 44 

56 

3 44 

56 

3-733 

57 

3 48 

117 

7 48 

x 77 

11 48 

237 

x 5 48 

297 

x 9 48 

357 

23 48 

57 

3 48 

57 

3.800 

58 

3 52 

I l8 

7 52 

1 78 

11 52 

238 

x 5 52 

298 

x 9 52 

358 

23 52 

58 

3 52 

58 

3.867 

59 

3 56 

119 

7 56 

179 

11 56 

239 

x 5 56 

299 

x 9 56 

359 

23 56 

59 

3 56 

59 

3 933 

60 

4 0 

120 

8 0 

180 

12 0 

240 

16 0 

100 

20 0 

360 

24 O 

60 

4 0 

60 

4.000 

_1 







1 




























































































































































































































































































































7 02 


TIME. 


Woodward in Smithsonian Geographical Tables, and they are 
reproduced here from that work. They are: 

1. By observing the transit of a star, whose right ascen¬ 
sion is known, across the meridian. 

2. By a single observed altitude of a star, which gives a 
fair approximate measure of time for geographic purposes. 

3. By equal altitudes of a star. 

The first is the most accurate and is that used in refined 
work. It is fully explained in Article 308 and in connection 
with determination of longitude (Chap. XXXV). The 
second method, elaborated in Article 306, requires a knowl¬ 
edge of the latitude of the place, which may be approximately 
measured from a good map or obtained by simple observa¬ 
tions (Art. 316). 

The third method is an extension of the preceding. A 
mean of the times when a star has the same altitude east and 
west of the meridian, is the time of meridian transit. With 
an engineer’s transit with telescope clamped, this method 
gives a good approximation to the time correction, freed of 
constant instrumental errors. The same method can be 
satisfactorily applied to the sun, using either engineer’s transit 
or sextant (Arts. 85 and 336). This is done by making 
measurements about two hours before and after noon of a 
series of altitudes before and after passage of the meridian, 
and taking a mean of the half-sums for time of meridian 
transit (Art. 308). 

306. Time by a Single Observed Altitude of a Star.— 

An approximate determination of time, often sufficient for 
the purposes of the geographer, may be had by observing the 
altitude or zenith distance of a known star. The method 
requires also a knowledge of the latitude of the place. 

Let z x = the observed zenith distance of the star; 
z — the true zenith distance of the star 

= *, + *. 


TIME BY MERIDIAN TRANSITS. 


70 S 

Then the hour-angle t may be computed by the formula 

. , 1, sin (<t — <t>) cos (cr — d) 

tan 2 ht = ----—-— - 1 ( T ^o\ 

cos cr cos (o’ — s) * ‘ 

in which cr = J(0+ d-f,?). 

307* Approximate Time from Sun. —To find approximate 
time or watch correction, observe the sun shortly before noon 
and, when it reaches its highest point, note the watch time of 
observed greatest altitude. This is a quick method, giving 
time within 10 m. 


Example of Computation , April 16 , 1898. 

The instant of sun’s greatest altitude occurs h. m. s. 

at apparent noon. 12 00 00 

Equation of time, April 16. — 17 

True local mean time. 11 59 43 

Watch time of instant of sun’s greatest 

altitude . 1 15 30 P.M. 

Watch correction. — 1 15 47 

Watch is therefore 1 h. 16 m. fast of local mean time, 
which is about what it would be if “Pacific standard” time 
were used in longitude 136°. 

308. Time by Meridian Transits. —The time of transit 
across the meridian being observed of a star whose right 
ascension is known, we have 

A T = a — T. .( I2 9) 

Meridian transits of stars may be observed by means of a 
theodolite or transit instrument mounted so that its telescope 
describes the meridian when rotated about its horizontal axis. 
The meridian transit instrument is specially designed for this 
purpose, and affords the most precise method of determining 
time. (Fig. 181.) 











704 


TIME. 


Since it is impossible to place the telescope of such an 
instrument exactly in the meridian, it is essential in precise 
work to determine certain constants, which define this defect 
of adjustment, along with the clock correction. These con¬ 
stants are the azimuth of the telescope when in the horizon, 
the inclination of the horizontal axis of the telescope, and the 
error of collimation of the telescope. 

Let a — azimuth constant, 

b — inclination or level constant; 
c =■ collimation constant. 

a is considered plus when the instrument points east of 
south ; b is plus when the west end of the rotation axis is the 
higher; and c is intrinsically plus when the star observed 
crosses the thread too soon from lack of collimation. 

Also let 

sin (0 — 6 ) 

A = -~— = the “azimuth factor”; (no) 

cos o ' J J 

cos (0 — 6 ) 

B — - -r — = the ‘‘ level factor ” ; . (ni) 

C = -5 = the “collimation factor.” . . (i?2) 

COS O \ j J 

Then, when a , b, c are not greater than io s each and 
preferably as small as I s each, 

T+ JT+Aa + Bb+Cc + r(T- T,j = a. . (133) 

This is known as Mayer s formula for the computation of 
time from star transits. 

The quantity Bb is generally observed directly with a 
striding-level. Assuming it to be known and combined with 
T , the above equation gives 

AT+Aa + Cc + r(T- T 0 ) = a- T. . . (134) 





TIME BY VARIOUS METHODS. 


70S 


This equation involves four unknown quantities , AT , a , 6', 
and r; so that in general it will be essential to observe at 
least four different stars in order to get the objective quan¬ 
tity AT. Where great precision is not needed, the effect of 
the rate, for short intervals of time, may be ignored, and the 
collimation c may be rendered insignificant by adjustment. 
Then the equation (134) is simplified into 

A 7 'fAa = a— T. .... (135) 

This shows that observations of two stars of different 
declinations will suffice to give AT. Since the factor A is 
plus for stars south of the zenith in north latitude and minus 
for stars north of the zenith, if stars be so chosen as to make 
the two values of A equal numerically but of opposite signs, 
AT will result from the mean of two equations of the form 
(435). With good instrumental adjustments, i.e. b and c small, 
this simple form of observation with a theodolite will give AT 
to the nearest second. 

A still better plan for approximate determination of time 
is to observe a pair of north and south stars as above, and then 
reverse the telescope and observe another pair similarly 
situated, since the remaining error of collimation will be partly 
if not wholly eliminated. Indeed, a well-selected and well- 
observed set of four stars will give the error of the timepiece 
used within a half second or less. This method is especially 
available to geographers who may desire such an approximate 
value of the timepiece correction for use in determining 
azimuth. It will suffice in the application of the method to 
set up the theodolite or transit in the vertical plane of Polaris, 
which is always close enough to the meridian. The deter¬ 
mination will then proceed according to the following pro¬ 
gramme : 

1. Observe time of transit of a star south of zenith, 

2. Observe time of transit of a star north of zenith. 


yo6 


TIME. 


Reverse the telescope and 

3. Observe time of transit of a star south of zenith; 

4. Observe time of transit of a star north of zenith. 

Each star observation will give an equation of the form 

(134), and the mean of the four resulting equations is 


2 A 2C 2 (T- T) 

AT+a -+ c- - + r— - oJ - 

4 ‘ 4 * 4 



(136) 


Now the coefficient of r in this equation may be always 
made zero by taking for the epoch T 0 the mean of the ob¬ 
served times T. Likewise, 2 A and 2C may be made small 
by suitably selected stars, since two of the A's and C’s are 
positive and two negative. The value £ 2 (a — T) is thus 
always a close approximation to AT (or the epoch T 0 = % 2 T, 
when 2 A and 2C approximate to zero. But if these sums 
are not small, approximate values of a and c may be found 
from the four equations of the form (134), neglecting the 
rate, and these substituted in the above formula will give all 
needful precision. 

For refined work, as in determining differences of longi¬ 
tude, several groups of stars are observed, half of them with 
the telescope in one position and half in the reverse position, 
and the quantities A T, a, c, and r are computed by the 
method of least squares (Art. 264). In such work it is 
always advantageous to select the stars with a view to mak¬ 
ing the sums of the azimuth and collimation coefficients 
approximate to zero, since this gives the highest precision 
and entails the simplest computations. 






CHAPTER XXXIII. 


AZIMUTH. 

309. Determination of Azimuth. —The azimuth of a line 
is the angle which it makes with a true north and south line; 
this angle is measured from the south around towards the 
west. It gives the initial direction from which the direc¬ 
tions of other lines in a trigonometric or traverse survey are 
derived. Azimuth is obtained by means of astronomic obser¬ 
vations by more or less approximate methods. For primary 
triangulation or traverse such observations are made by the 
most accurate methods, and at intervals not greater than 50 
to 75 miles in primary triangulation, and not exceeding 10 
miles in primary traverse. 

The determination of the azimuth of a terrestrial line 
consists in the measurement of an angle between two vertical 
planes, one passing through a terrestrial mark called the azi¬ 
muth mark and the center of the instrument, and the other 
through an observed star and the center of the instrument. 
The exact time at which pointing is made upon the star must 
be noted by a chronometer, the error of which is known, be¬ 
cause the angle of the star between these two points is con¬ 
tinually changing. With the aid of the recorded time, the 
hour-angle of the star and its azimuth as seen from the station 
may be computed. The measured hour-angle at the station 
between the star and the azimuth mark added to or subtracted 
from the computed azimuth of the star gives the azimuth of 
the terrestrial mark from the station. 

310. Observing for Azimuth. —The instrument used in 

making azimuth observations is a theodolite similar to that 

707 


AZIMUTH. 


708 

used in primary triangulation or traverse (Art. 241). Azi¬ 
muth observations may be made, however, for secondary 
purposes, as for the reduction of transit traverse lines by 
means of latitudes and departures with the instrument used 
in running the traverse (Arts. 90 and 85). In this case the 
method employed is similar to that hereafter described, but 
is more simple because the instruments employed are less 
accurate and call, therefore, for less care in their use (Art. 
311). The observation for azimuth by astronomic methods, 
as those required in the determination of primary azimuths of 
a base line or in a belt of triangulation, consists in the meas¬ 
urement of the horizontal angle between some close circum¬ 
polar star, usually Polaris, and a terrestrial mark. The latter 
is generally a bull’s-eye lantern set at a distance of at least 
half to one mile from the observer’s station. 

Since the star is at a much higher angle than the terres¬ 
trial mark, it is necessary to measure the error of level and to 
correct for it in addition to carefully leveling the instrument. 
As a result the value of a division of the level-bubble must be 
accurately known. Observations for azimuth may be made 
at any time of night, preferably near the time of elongation, 
since the star is then moving most slowly in azimuth and any 
error in time has the least effect in the result. The error of 
the watch must be obtained by comparison with a standard of 
time and corrected for the difference in longitude between the 
observing station and the meridian of such standard of time. 

311. Approximate Solar Azimuth. —This observation 
may be made to obtain meridians in public-land surveys and 
for similar approximate work, the only instrument required 
being an engineer’s transit in good adjustment, thus doing 
away with the solar (Art. 339) or other special attachments. 

Observations should be made in the morning and afternoon, 
the routine being as follows: 

1. Point on azimuth mark and read horizontal circle; 

2. Point on sun with telescope direct; 


A P PR OX/A/A PE SOLAR A ZIM U PH . 


709 


3. Point on sun with telescope inverted; 

4. Point on azimuth mark and read horizontal circle. 

To eliminate errors of collimation and- of verticality and 
horizontality of cross-hairs, two pointings on the sun are made 
thus: 

In the morning observations the sun is passing to the 
right and rising. Observe in upper left and lower right 
quadrants, the contacts being on lower and first limb. This 
operation is so performed as to facilitate manipulation of tan¬ 
gent motions of the instrument and in order that only one 
slow motion shall be used during the observation. The hori¬ 
zontal thread in the first case should be set above the limb of the 


sun so that, as the latter rises, it may be closely watched, and 
just as the sun moves off contact the vertical cross-hair must 
be made tangent to the first limb by the slow motion. In 
the second case the vertical cross-hair may be set just to the 
right of the limb and slow motion made with the vertical 
tangent screw. In the afternoon the routine is to observe in 
the upper right and lower left quadrants, the method of 
manipulation being similar to the above. 

Let 5“ = sun’s diameter; 
c = collimation error; 

0 =l position of azimuth mark; 

a' = A' — s — c — horizontal angle between some 

fixed point, 0 , and the sun’s 
center in the first observation 
with telescope; 

a " = A " -f - s — c — horizontal angle between some 

point, 0 , and the sun’s center on 
the second observation with the 
telescope inverted; 

A ' = circle reading when telescope is pointed at sun’s 
first limb; 

A" = circle reading when telescope is pointed at sun’s 
second limb. 


Then 


(<T -f 6 ") 


i(A' + A'). 


2 


(• 37 ) 



7io 


AZIMUTH . 


H’ = A' +J + *; 

H" = h" — s — c f 

H + H" __ h' + h" 

2 ~~ 2 


Let d = declination = \{Ji + 0 + f) ; 

^ = altitude corrected for refraction; 
0 = latitude; 

p — sun’s or star's polar distance; 
a — azimuth counted from the north. 


Then 


1 

tan —a — 

2 


sin (s — h) sin (s — 0) 
cos . cos (s — p) 


(138) 


EXAMPLE. 


Azimuth Mark. 
Ver. A. Ver. B. 

25° 24' 205° 23 

25 24 205 25 


Sun, Horizontal Circle. 
Ver. A. Ver. B. 

23 ° 02' 203° OI 

23 14 203 15 


Mean, 25° 24' 23 0 08' 

25 0 24' 
23 08 

Angle between mark and mean place of sun ... .. 2 0 16' 

Sun, Vertical Circle. Index. 

8 ° 37 ' +3 

12 47 - 1 


io° 42' + 1 

+ 1 

Altitude ...... .. io° 41' 

Refraction. — 5 always negative 


Corrected altitude.. io G 36' 


More pointings than two are desirable, and these should be 














APPROXIMATE SOLAR AZIMUTH. 


7 ii 


equally divided between the two limbs so as to eliminate the 
semi-diameter of sun. 

First find sun s polar distance corresponding to approxi- 
mate r time of observation. 


Date, April 16, 1898. 

Approximate time of observation by watch. 7 h. 15 m. A.M. 

Watch correction (Art. 303). _ x I5 


00 


Map longitude from Greenwich 136° = hrs.= -f 8 48 always positive 

Greenwich time, approximate. 15 hours after midnight 

Subtract 12 “ 

Greenwich time after noon, because declination 

is given for noon. 3 P>M . 

Hourly change in declination, April 16. -j- 53" 

3 h. X 53 " = + 159 " = + 2' 39" 

Sun’s declination, April 16, Greenwich noon 

(Almanac). 

Sun’s declination, April 16, 3 hours later. 

Sun’s polar distance 90° — sun’s declination.... 

Sun’s polar distance. 79 0 44' 

Latitude. 64 23 

Observed altitude... 10 36 


+ 10 13 49 
10 16 28 
79 44 


Sum. 154 43 

i sum . 77 21 log. sec. 

5 sum — altitude. 66 45 “ sin. 

\ sum — latitude. 12 58 “ sin. 

4 sum — polar distance. 2 23 “ sec. 

Check sum. 154 41 

Log. tan. 4 azimuth. 

4 azimuth. 44 0 09' 

Azimuth. 88 18 

Angle between sun and mark. 2 16 

Azimuth of mark (east of north). 90° 34' 


= 0.6596 
= 9-9632 

= 9 35 io 
= 0.0004 

2 ) 9-9742 

9.9871 

[from north, 
always measured 


Another and quite simple formula for determining the 
meridian by a single solar observation is given by Mr. W. 
Newbrough and is as follows: 


Cos — 



cos 5 cos (S — p) 
cos (/) cos h 


( r 39 ) 


where a = sun’s azimuth measured from north; 
p — sun’s polar distance or codeclination; 
h -- sun’s altitude minus correction for refraction ; 

































712 


AZIMUTH. 


S — half the sum of polar distance, latitude, and true 
azimuth; and 

0 = latitude of place of observation. 

312. Azimuths of Secondary Accuracy. —In observing 
an azimuth on a primary traverse, less care and accuracy are 
required than in observing a primary azimuth in triangulation. 
The following procedure illustrates the observing and com¬ 
puting of such a secondary azimuth. The instrument is cen¬ 
tered over any station on the traverse line, and the azimuth 
mark placed at the next station. This should be at least 1500 
feet from the instrument, and may be a narrow slit in a box 
containing a light, or the small colored light on the side of a 
bicycle-lamp carefully centered over the station. The error 
of a good watch should be known by comparison with 
telegraph time signal, which is sent over all Western Union 
lines once every day. 

The angle is then measured between the azimuth mark 
and Polaris, making at least six pointings at both mark and 
star, three with telescope direct and three with telescope re¬ 
versed, and at each star pointing the time is noted to nearest 
second. Then the reduction is made in the following manner: 

EXAMPLE. 

Latitude 34 0 32' (to nearest minute); longitude 92 0 35'. 

(Geo. T. Hawkins, Observer and Computer.) 

Azimuth observation at Benton, Ark., October 8. 1898. Between in¬ 
strument traverse stations 106 and 107. Instrument at sta. 107, mark at sta. 
106. 90° meridian time by watch, which was compared with Western Union 


time at 10 o’clock to-day an 

d found to be 

1.0 minute slow (Arts. 23oand 234). 

Time by watch. 

7 h 

io m 

49 ’ 

= mean time of 6 pointings on Polaris. 

Correction of watch. 

+ 

OT 

OO 



7 

II 

49 

= 90° meridian time of observation. 

Correction for longitude. 

— 

IO 

20 



7 

OI 

29 

= astr. local mean time observation. 

Upper culmination, subtract... — 

12 

14 

06 


Hour-angle Polaris at observation 

18 

47 

23 

U. C. Polaris Oct. 1 (Table LXIII). 12 37.7 

Subtract from . 

23 

56 

06 

Reduction to Oct. 7 (Table LXIV).. — 23.6 


Time argument ., .. 5 08 43 U. C. Polaris, Oct. 7. TP14T1 

Azimuth of Polaris at observation (Table LXV). i° 28' 17" or 181 0 28' 17" 

Angle at sta. 106 bet. sta. 107 and Polaris (mean of 6 readings). +43 02 00 


Azimuth from sta. 107 to sta. 106. = 224° 30' 17" 

- 180° 


Azimuth from sta. 106 to sta. 107 


44 ° 30' 17" 



















TIMES OF CULMINATIONS AND ELONGATIONS. 713 

The annexed diagram (Fig. 180a) will show in their proper 
relation the various aspects of Polaris in its daily apparent motion 
around the north-polar point. 

This must be carefully studied, as the illustration of Table LX, 
for finding at any hour the hour-angle and azimuth of Polaris, 



and the resulting meridian, at times when more direct methods 
are not available. 

In Fig. i8otf the full vertical line represents a portion of 
the meridian passing through the Zenith Z (the point directly 
overhead), and intersecting the northern horizon at the north 
point N, from which, for surveying purposes, the azimuths of 












7 l 4 


AZIMUTH. 


Polaris are reckoned east or west. The meridian is pointed 
out by the plumb-line when it is in the same plane with the eye 
of the observer and Polaris on the meridian, and a visual repre¬ 
sentation is also seen in the vertical wire of the transit, when it 
covers the star on the meridian. 

When Polaris crosses the meridian it is said to culminate; above 
the pole (at S) the passage is called the upper -culmination, in 
contradistinction to the lower culmination (at 5 '). 

In the diagram—which the surveyor may better understand 
by holding it up perpendicular to the line of sight when he looks 
toward the pole—Polaris is supposed to be on the meridian, 
where it will be about noon on April io of each year. The star 
appears to revolve around the pole, in the direction of the arrows, 
once in every 23 11 56™. 1 of mean polar time; it consequently 
comes to and crosses the meridian, or culminates, nearly four 
minutes earlier each successive day. The apparent motion of 
the star being uniform, one quarter of the circle will (omitting 
fractions) be described in 5 h 59™, one half in n h 58™, and three 
quarters in 57“. For the positions s 1, s 2 , s 3 , etc., the angles 
SPsi , SPs 2 , SPs 3 , etc., are called hour-angles of Polaris, for the 
instant the star is at $1, s 2 , or s 3 , etc., and they are measured 
by the arcs Ss 1, Ss 2 , Ss 3 , etc., expressed (in these instructions) in 
mean solar (common clock) time, and are always counted from the 
upper meridian (at 5 ), to the west, around the circle from o h o m 
to 23 11 56™. 1, and may have any value between the limits named. 
The hour-angles, measured by the arcs Ss u Ss 2 , Ss 3 , Ss 4 , Ss 5 , and 
Ss 6 , are approximately i h 8 m , 5 h 55™, 9 h q m , iq h 52™, i8 h oi m , 
and 22 h 48 111 , respectively; their extent is also indicated graphic¬ 
ally by broken fractional circles about the pole. 

Suppose the star observed at the point S 3 ; the time it was 
at 5 (the time of upper culmination), taken from the time of 
observation, will leave the "arc Ss 3 , or the hour-angle at tho 
instant of observation; similar relations will obtain when the 
star is observed in any other position; therefore, in general * 

Subtract the time oj upper culmination from the correct local 


TIMES OF CULMINATIONS AND ELONGATIONS. 7 I 5 


mean time oj observation; the remainder will be the hour-angle 
oj Polaris expressed in time, or the “argument jor Table LXIV .” 

The observation may be made at any instant when Polaris 
is visible, the exact time being carefully noted. 


Table LX. 

(From Magnetic Declination Tables, U. S. Coast and Geodetic Survey. Com¬ 
puted for latitude 40° north and longitude 90° or 6 h west of Greenwich.) 


Date. 

East 

Elongation. 

Upper Cul¬ 
mination. 

West 

Elongation. 

Lower Cul¬ 
mination. 

1902 

h 

m 

h 

in 

h 

m 

h 

m 

January 1 . * . 

0 

45-8 

6 

40.6 

12 

35-3 

18 

38-7 

January 15. 

2 3 

46.6 

5 

45-3 

II 

40.0 

17 

43-4 

February 1 . 

22 

39-5 

4 

38.2 

10 

3 2 -9 

16 

3 6 -3 

February 15. 

21 

44.2 

3 

42.9 

9 

37-7 

15 

41.0 

March 1 . 

20 

49.0 

2 

47-7 

8 

42.4 

14 

45-8 

March 15. 

19 

54 -o 

1 

5 2 -7 

7 

47-3 

13 

50-7 

April 1 .... . ... 

18 

47.0 

0 

45-6 

6 

40-3 

12 

43-7 

April 15. 

i 7 

52.0 

2 3 

46.7 

5 

45-3 

II 

48.6 

May 1 . 

16 

49.1 

22 

43 - 8 

4 

42.5 

IO 

45-7 

May 15. 

i 5 

54-2 

21 

48.9 

3 

47.6 

9 

50.8 

June 1 . 

14 

47-5 

20 

42.3 

2 

40.9 

8 

44.2 

June 15. 

13 

52.6 

19 

47-4 

1 

46.0 

7 

49-3 

July . . 

12 

5 °.° 

18 

44.8 

o- 

43-4 

6 

46.7 

July 15 . 

11 

55 • 1 

1 7 

49.9 

23 

44.6 

5 

51.8 

August . . 

10 

48.6 

16 

43-4 

22 

38.0 

4 

45-3 

August 15. 

9 

53-7 

15 

48.5 

21 

43 -i 

3 

5°-4 

September .. 

8 

47.1 

14 

41.9 

20 

3 6 -5 

2 

43-8 

September 15 . 

7 

52.2 

13 

47.0 

19 

41 . 6 

1 

48.9 

October .. 

6 

49-3 

12 

44.1 

18 

3 8 -7 

0 

46.0 

October 15 . 

5 

54-3 

11 

49.1 

O 

43-7 

2 3 

47 - 2 

November .. 

4 

47-5 

10 

42.3 

16 

3 6 -9 

22 

40.4 

November 15. 

3 

5 2 -3 

9 

47.1 

15 

41.8 

21 

45 - 2 

December .. 

2 

49-3 

8 

44.1 

14 

38.8 

20 

42.2 

December 15. 

1 

54 -o 

7 

48.8 

13 

43- 6 

T -9 

46.9 








































AZIMUTH. 



A. To refer the above tabular quantities to years subsequent to 

IQ02 : 


1903 

add 

1.4 

minutes. » 


f add 

2 .8 

“ up to-March 1 

1904 

\ subtract 

I. I 

“ on and after March 1 

1905 

add 

0.2 

< c 

1906 

<< 

1-5 

i c 

1907 

a 

2.9 

i i 

1908 

“ / 

4.2 

“ up to March 1 

i 

o -3 

‘ ‘ on and after March 1 

1909 

c i 

i -7 

6 i 

1910 

i c 

3 -o 



B. To refer to any calendar day other than the first and fifteenth 
of each month: Subtract the quantities below from the tabular 
quantity for the preceding date. 


Table LX'f. 


Day of Month. 

Minutes. 

Number 
ot Days 
Biapsed. 

Day of Month. 

Minutes. 

Number 
of Days 
Elapsed. 

2 or 

16 

3-9 

1 

10 or 

24 

35-5 

9 

3 

17 

7-9 

2 

11 

25 

39-4 

10 

4 

18 

11.8 

3 

12 

26 

43-3 

11 

5 

19 

15.8 

4 

13 

27 

47-3 

12 

6 

20 

19.7 

5 

14 

28 

51.2 

13 

7 

21 

23.6 

6 


29 

55-2 

14 

8 

22 

27.6 

7 


30 

59 -i 

i 5 

9 

23 

3 i -5 

8 


3 i 

63.0 

16 


C. To refer the table to standard time: Add to the tabular 
quantities four minutes for every degree of longitude the place 
is west of the standard meridian, and subtract when the place 
is east of the standard meridian. 















TIMES OF CU.LMINA TIONS AND ELONGA T10NS. 7 17 

Table LXII. 


AZIMUTHS OF POLARIS AT ELONGATION. 
Between 1900 and 1910 and Latitudes 25 0 and 75 0 North. 
(From U. S. Land Survey Manual.) 


Latitude. 

1905. 

1906. 

1907. 

1908. 

1909. 

1910. 

O 

O 

/ 

O 

• 

O 

/ 

O 

r 

0 / 

0 

/ 

30 

1 

23.1 

1 

22.8 

1 

22.4 

1 

22.1 

1 21.7 

1 

21.3 

31 


24.0 


23.6 


23.2 


22.9 

22.5 


22.2 

3 2 


24.9 


24-5 


24.1 


23.8 

2 3-4 


23.1 

33 


2 5 -9 


2 5-5 


25-i 


24.7 

24-3 


24.0 

34 


26.9 


26.5 


26.1 


25-7 

2 5-3 


25.0 

35 

1 

27.9 

1 

2 7 • 5 

1 

27.1 

1 

26.8 

1 26.4 

1 

26.0 

36 


29.0 


28.6 


28.2 


27.9 

2 7-5 


27.1 

37 


30.1 


29.7 


2 9-3 


29.0 

28.6 


28.2 

38 


3 i -4 


31.0 


30.6 


30.2 

29.8 


29.4 

39 


3 2 • 7 


3 2 -3 


31.8 


31-4 

31.0 


3 °.6 

40 

1 

34.0. 

1 

33-6 

1 

33-2 

1 

32.8 

1 32-4 

1 

32.0 

41 

42 


35 - 4 

3 6 - 9 


35 - o 

3 6 - 5 


34-6 

36.0 


34 - 2 

35 - 6 

33-8 

35-2 


33 - 4 

34 - 8 

43 


38.5 


38.1 


37 - 6 


37-2 

36.8 


36.3 

44 


40.1 


39-7 


39-2 


38.8 

38-4 


37-9 

45 

1 

41.8 

1 

41.4 

1 

40.9 

1 

40.5 

1 40.1 

1 

39-6 

46 


43-7 


43 - 2 


42. ) 


42.3 

41.9 


41.4 

47 


45 - 6 


45 - 1 


44.6 


44.2 

43-7 


43-3 

48 


47-7 


47.2 


46.7 


46.3 

45-8 


45-3 

49 


49.8 


49-3 


48.8 


48.4 

47-9 


47-4 

50 

1 

5 2 -° 

1 

5 i -5 

1 

5 1 -° 

1 

5°. 6 

1.5°. 1 

1 

49.6 


Table LXIII. 

CORRECTION TO AZIMUTHS OF POLARIS FOR EACH MONTH. 

(From U. S. Land Survey Manual.) 


For Middle of— 

Latitude. 

For Middle of— 

Latitude. 

25 °. 

40°. 

55 °. 

25 °. 

40°. 

55 °. 

January. 

February. 

A/TnrrVi. 

t 

-o -3 
-o -3 
— 0.1 

r 

-0.4 
-o -3 
— 0.2 

f 

-o -5 

-0.4 

— 0.2 

July . 

August. 

September. 

r 

+ 0.2 
+ 0.1 
0.0 

f 

+ 0 -3 
+ 0.1 
— 0.1 

/ 

+ 0.4 
+ 0.2 
— O . I 

A nri 1 

0.0 

0.0 

0.0 

October. 

— 0.2 

-o *3 

— 0.6 

-O.4 

-O.7 

-O.9 

TVT a v 

+ 0.2 
+ 0.2 

+ 0.2 
+ 0.3 

-)- 0.2 
+ 0.4 

November. 

- 0-5 
— 0.6 

June . 

December. 

-0.8 






























































718 Table LXIV. 

(From U. S. Land Survey Manual. The hour-angles are expressed in mean solar time. The 


is o m .5 greater 


Star and Azimuth. 

W. oi N. when hour-angle is less 
than u h 58“. 

E. of N. when hour-angle is greater 
than n h 58™. 

Time argument, the star’s hour-angle 
(or 23 h 56""*. 1 minus the star’s 
hour-angle), for the year— 

Hours. 

O 

O 

M 

1 n 
0 
0 

M 

0 

0 

O' 

M 

O 

a 

w 

00 

0 

O' 

M 

O' 

0 

O' 

w 

1910. 

1911. 

h. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

0 

O 

O 

0 

O 

O 

O 

O 

O 


5 

5 

5 

5 

5 

5 

5 

5 


9 - 

9 - 

9 • 

9 - 

9 - 

IO 

IO 

IO 


14. 

14. 

14. 

14. 

14. 

14- 

14. 

1.5 


19 

19 

19 

19. 

19. 

19. 

19. 

19 . 


24 

24 

24 

24 

24. 

24. 

24. 

24. 


28. 

29 

29 

29 

29 

29. 

29. 

29. 


33 - 

33 - 

34 

34 

34 

34 - 

34 - 

34 - 


38 . 

38 . 

38 . 

39 

39 

39 

39 - 

39 - 


43 

43 - 

43 - 

44 

44 

44 

44 - 

44 - 


48 

48 

48. 

48. 

49 

49 

49 - 

49 - 


53 

53 

53 - 

53 - 

54 

54 

54 - 



? 8 

f 8 

,58 • 

,5 8 • 

,59 

59 


0 l 

1 

3 

3 

3 * 

3 - 

4 

4 - 

4 - 

5 \ 


7 • 

8 

8. 

8. 

9 

9 - 

9 - 

10 


13 

13 

13. 

14 

14 

14. 

15 

15 


18 

18 

18. 

19 

19. 

19. 

20 

20 . 


23 

23. 

23 - 

24 

24. 

25 

25. 

26 


28 

28. 

29 

29. 

29. 

30 

30. 

31 


33 - 

33 - 

34 

34 - 

35 

35 - 

36 

36. 


38. 

39 

39 - 

40 

40. 

4 i 

41. 

42 


44 

44. 

45 

45 • 

46 

46. 

47 

47 • 


49 

50 

50 - 

51 

5 i - 

52 

52. 

53 


I? 4 ' 

,55 

51 . 

56. 

57 

57 • 

58 

58. 

2 

O 

0. 

1. 

2 

2 . 

3 

4 

4 • 


6 

6. 

7 

7. 

8. 

9 

9. 

10. 


XI . 

12 

12. 

13. 

14 

15 

15. 

16. 


17 

18 

18. 

19. 

20 

21 

21. 

22. 


23 

24 

24. 

25. 

26 

27 

28 

28. 


29 

30 

30. 

31 - 

32. 

33 

34 

35 5 


35 

36 

37 

38 

38 . 

39 - 

40. 

41. 5 


41 - 

42. 

43 - 

44 - 

45 

46 

47 

48 5 


48 

49 

50 

5 i 

52 

53 

54 

.5.5 5 




$ 6 ‘. 

f 7 ’ 

)> 8 • 

if?’ 1 

I 

2 5 

3 

1. 

2. 

3 - 

4 - 

6 

7 

8 

9 6 


8. 

IO 

I I 

I 2 

13 

14- 

15 - 

17 C 


16 

17 

18. 

19. 

21 

22 

23 - 

25 ( 


23. 

25 

26 

27 • 

29 

3 °- 

3 i - 

33 6 


31 - 

33 

34 - 

35 - 

37 

38 . 

40. 

42 6 


39 - 

41 

43 

44. 

46 

47 - 

49. 

S rw 6 


48. 

50 

if 2 

, 53 - 

55 

5 7 

50 

0. 7 


if 8 

??• 

1 . 

3 - 

5. 

7 - 

9 - 

11 . 7 

4 

8 

IO 

1 2. 

14. 

16. 

19 

21 

23. 7 


19. 

22 

24 

26. 

29 

32 

34 - 

37 - 7 


32 

34 - 

37 - 

40. 

43. 

46. 

5 ° 

53 - 7 


46. 

, 5 ° 


57 J 

2 

6. 

11 

16. 7 

5 1 

5 

TO 

l6 

23.I32 

42.« 


8 


40. 



1 



8 


Polaris above the Pole. 

To determine the true meridian, the azimuth 
will be laid off to the east when the hour- 
angle is less than u h 58™, and to the west 
when greater than n h 58™. 


Azimuths for latitude— 


O 

30 


o 

32 


o 

34 


o 

2 

3 
5 

7 

9 

10 

1 2, 

14 

16 

17 . 


21 


|23 


o 

2 

3 - 

5 . 

7 
9 
11 
1 2. 

14. 

16 

18 

20 

21. 

23. 

25. 

27 

29 

3 i 

32 . 

34 - 

36 

38 
39 - 
4 i • 

43 

45 

47 

48. 

50 . 

52 

54 

56 

57 . 


o 

2 

3 - 

5. 

7. 

9 
11 


o 

36 


o 

2 

4 

5 - 

I 7 • 
I 9 ' 

11, 


13 13 
i4- 15 


16 . 
18. 
20. 

22 

24 

26 

27 • 

29. 
3 i • 
33 

35 

37 

38 . 

40. 

42 . 

44 

46 

48 

49 - 


17 . 

19 

21 
22. 
24. 

26. 

28 

30 

32 

34 

36 

38 

39 

4 i 

43 

45 . 

47 

49 

5 i 


51 . 53 


65 

66 . 

68 . 

70. 

72 


77-79 


83 

85 


53 - 

55 - 

57 

59 ^ 

61 

63 

64. 

66 . 

68 . 

70 

72 

74 

76 

77 - 
79 - 

81 . 
83 
85 
87 


55 
56 . 
58 . 
60. 
62. 

64. 

66 

68 

70 

72 

74 

75 - 

77 - 

79 - 
81 . 

83. 

85 

87 

'80 


O 

O 

O 

O 

38 

40 

42 

44 

• 

r 

/ 

* 

O 

0 

O 

O 

2 

2 

2 

2 

4 

4 

4 

4 - 

6 

6 

6 

6. 

8 

8 

8. 

8. 

9 - 

10 

10. 

10. 

I I . 

I 2 

I 2 . 

13 

13 - 

14 

14- 

15 

15 - 

16 

l6 . 

17 

17 - 

18 

18. 

19. 

19. 

20 

20 . 

21 . 

21 . 

22 

22. 

23 - 

23. 

24 

25 

26 

25 

26 

27 

28 

27 

28 

29 

30 

29 

30 

31 

32 

31 

32 

33 

34 - 

33 

34 

35 

36 . 

35 

36 

37 

38 . 

37 

38 

39 - 

41 

39 

40 

41. 

43 

40. 

42 

43 - 

45 

42. 

44 

45 - 

47 

44 - 

46 

47 - 

49 

46. 

48 

49 - 

51 . 

48. 

50 

5 i - 

53 - 

50 . 

52 

53 - 

55 - 

52 . 

54 

5 6 , 

-aJ 

54 - 

56 

58 

60 



56. 

58 

60 

62 

5 jL,S 6 ° 

62 

64 

60 

62 

64 

66 

62 

64 

66 

68. 

64 

66 

68 

70. 

66 

68 

70 

72. 

68 

70 

72 

74 - 

70 

72 

74 - 

77 

72 

74 

76. 

79 

74 

76 

78 . 

81 

76 

78 

80. 

83 

77 • 

80 

82. 

85 - 

79 - 

82 

84. 

87. 

81. 

84 

86. 

89. 

83. 

86 

88. 

91. 

85 - 

88 

90. 

94 

87. 

90 

93 

96 

89. 

92 

95 

98 

91. 

94 

97 

IOO 



o 

46 


o 

2 

4 

6 

9 
11 

13. 

IS- 

18 

20 
22. 

24. 

27 

29 

31 - 

33 - 

36 

38 

40 


O 

48 

O 

50 


S 

0 

0 

2 . 

2 . 

4 - 

5 

7 

7 

9 

9 - 

I I . 

I 2 

14 

14. 

16 

17 

18. 

19 

21 

21. 

23 

24 

25. 

26. 

28 

29 

30 

31. 

32. 

34 

35 

36. 

37 • 

39 

39 - 

41 • 

42 

43. 

44 

46 

46 . 

48. 

48. 

50. 

51 

53 

53 

55 

55 - 


—-j 

1 6° 

60 

62 . 

62 

64. 

64. 

67 

66. 

69. 

69 

72 

7 i • 

74 - 

73 - 

76 . 

76 

79 

78 

81. 

80. 

84 

82. 

86 

85 

88. 

87 

91 

89. 

93 - 

91. 

95 - 

94 

98 

96 

IOO . 

98. 

103 

101 

105 

103 

107. 

i° 5 - 

110 

107 . 

r 1 2 




































































































































Azimuths of Polaris. 71 8a 


occurrence of a period after minutes of time or of an hour-angle indicates that its value 
than printed.) 


Star and Azimuth. 

W. of N. when hour-angle is less 
than 11 h 58"*. 

E. of N. when hour-angle is greater 
than n h 58 m . 

Time argument, the star s hour-angle 
(or 23 h s6 m .i minus the star’s 
hour-angle), for the year— 

To 

Polaris the Pole. 

determine the true meridian, the azi¬ 
muth will be laid off to the east when the 
hour-angle is less than n h s8 m , and to the 
west when greater than n h s8 m . 

Hours. 

Tfr 

O 

O 

M 

in 

0 

Q\ 

M 

<© 

O 

o 

w 

1 

1907. 

00 

0 

o> 

w 

0 

0 

0 

M 

1910. 

M 

M 

W 

Azimuths for latitude— 

O 

30 

O 

32 

0 

34 

0 

36 

O 

38 

O 

40 

0 

42 

O 

44 

O 

46 

O 

48 

O 

50 

h. 

m. 

1 







• 


/ 

/ 

/ 

/ 

/ 

f 

/ 

/ 

r 

6 

9 - 

m. 

m. 

m. 

m. 

m. 



83 

85 

87 

89 

91 . 

94 

97 

IOO 

103. 

107. 

112 


4,5 

j °. 

34 

27 

18 

8 

m. 

m. 

81. 

83 

85 

87 

89. 

92 

95 

98 

101. 

105. 

log . 

7 

4 

°-! 

156 . 

52. 

48. 

44 

39 

34 

79 - 

81. 

83 

85 

87. 

90 

93 

96 

99 - 

103 

107 


18. 

16 

13 

10 

7 

4 

o.| 


78 

79 - 

81. 

83. 

85. 

88 

90. 

93 - 

97 

IOO . 

104. 


3 i • 

29 

26. 

24 

21. 

19 

l6 

13 

76 

77 • 

79. 

81. 

83. 

86 

88. 

91. 

95 

98. 

102 


42, 

40. 

38. 

36. 

34 - 

32 

29. 

27 

74 - 

76 

77. 

79 - 

81. 

84 

86. 

89. 

92. 

96 

IOO 


53 

51 • 

49 . 

47 - 

45 - 

43 - 

41 • 

39 - 

72 . 

74 

76 

77 • 

79 - 

82 

84. 

87. 

90. 

94 

97. 

T" 

2. 

1 

59 - 

57 . 

56 

54 

52 . 

50. 

71 

72 . 

74 

76 

77 - 

80 

82. 

85. 

88. 

91 . 

95 


II. 

10 

8. 

7 

5 - 

4 

2 . 

I 

69 

70. 

72 

74 

75 - 

78 

80. 

83 

86 

89 

92. 


20 

18. 

17 - 

16 

14. 

13 

11. 

10 

67. 

68. 

70. 

72 

74 

76 

78. 

81 

84 

87 

90 


28 

27 

25 • 

24. 

23 

21. 

20. 

19 

65. 

67 

68. 

70 

72 

74 

76. 

79 

81. 

84. 

88 


36 

35 

33 - 

32. 

31 

30 

28. 

27. 

64 

65 

66. 

68 

70 

72 

74 - 

7.7 

79 - 

82 . 

85. 


43 - 

42. 

41 . 

40 

39 

38 

36. 

35 • 

62 

63. 

64. 

66. 

68 

70 

72 

74 - 

77 - 

80 

83 


50 . 

49. 

48. 

47 - 

46. 

45 - 

44 - 

43 

60. 

61. 

63 

64. 

66 

68 

70 

72 . 

75 

78 

80. 


57 - 

56. 

55 - 

54 - 

53 - 

52 . 

51 • 

50 . 

58 . 

59.1 

61 

62. 

64 

66 

68 

70. 

73 

75 - 

78 

9 

4 - 

3 - 

2. 

1 . 

1 

0 

59 

58 

57 

58 

59 

60. 

62. 

64 

66 

68 

70. 

73 

76 


11 

10. 

9 - 

8. 

7 - 

6. 

5 - 

5 

55 

56 

57 - 

59 

6^ 

62 

64 

66 

68. 

7 i 

73 - 


1 7 • 

17 

16 

15 

14. 

13. 

1 2 

11. 

53 - 

54 - 

55 - 

57 

58. 1 


62 

64 

66 

68. 

7 i 


24 

23 

22 . 

21 . 

20. 

20 

19 

18 

5 i • 

52 . 

53 - 

55 

56. 

MM 

58 1 

60 

62 

64 

66. 

68. 


3 ° 

20. 

28. 

28 

27 

26. 

25 • 

24. 

49 - 

50 . 

52 

53 

54 - 

56 

58 

sp i 

62 

64 

66. 


36 

35 • 

35 

34 

33 - 

32. 

32 

3 i 

48 

49 

50 

5 i 

52 . 

54 

55 - 

57. 

59 - 

61. 

64 


42 

41 . 

41 

40 

39 - 

38 . 

38 

37 • 

46 

47 

48 

49 - 

50 . 

52 

53 - 

55 - 

57 - 

59 - 

mb' 


48 

47 • 

47 

46 

45 • 

45 

44 

43 - 

44. 

45 - 

46. 

47 - 

48. 

50 

5 i • 

53 - 

55 

57 

59 


54 

53 

52. 

52 

5 i- 

5 i 

50 

49. 

42. 

43 - 

44 - 

45 - 

46. 

48 

49 - 

5 i 

53 

55 

57 


59 . 

59 

58. 

57 - 

57 

56 . 

, 5 6 

55 - 

4 i 

41 . 

42. 

43 - 

44 - 

46 

47 - 

49 

50 . 

52 . 

54 - 

10 

5 

4 - 

4 

3 - 

3 

2 . 

2 

I . 

39 

40 

40. 

41 . 

43 

44 

45 - 

47 

48. 

50J 

52 


10 . 

10 

9 • 

9 

8. 

8 

7 - 

7 

37 . 

38 

39 

40 

4 i 

42 

43 - 

45 

46. 

48 

49 


16 

15 • 

15 

14. 

14 

13 - 

13 

1 2 . 

35 - 

36 

37 

38 

39 

40 

4 i 

42. 

44 

45 - 

47 - 


21. 

21 

20. 

20 

19. 

19. 

19 

18. 

34 

34 - 

35 

36 

37 

38 

39 

40. 

42 

43 - 

45 


27 

26. 

26 

25 - 

25 

25 

24. 

24 

32 

32 . 

33 - 

34 

35 

36 

37 

38 . 

39 - 

41 

42 . 


32 

32 

31 • 

3 i 

30 - 

30 

30 

29. 

3 ° 

3 i 

3 i • 

32 

33 

34 

35 

30 

37 - 

39 ] 

40 


37 ■ 

37 

36. 

36. 

36 

35 • 

35 

35 

28 . 

29 

29. 

3 °- 

3 i 

32 

33 

34 

35 • 

36. 

38 


42 . 

42. 

42 

41 . 

41 . 

4 i 

40. 

40 

26 . 

27 

28 

28 . 

29 

30 

3 i 

32 

33 

34 

35 - 


48 

47 • 

47 • 

47 

46. 

46. 

46 

45 - 

25 

25 • 

26 

26. 

27 

28 

29 

30 

31 

32 

33 


53 

52 . 

52. 

52 

52 

5 i • 

5 i 

5 i 

23 

23 - 

24 

24. 

25 • 

26 

27 

27 • 

28. 

29 - 

31 


58 

58 


§7 • 



1 s6 ' 

56 

21 . 

22 

22 

22 . 

23 

24 

25 

25 - 

26 . 

27 • 

28. 

11 

3 

3 

2 . 

2 . 

2 

2 

I . 

I . 

19. 

20 

20 . 

21 

21 . 

22 

22 . 

23 - 

24 

25 

26 


8. 

8 

8 

7 . 

7 

7 

7 

6. 

18 

18 

18. 

19 

19. 

20 

20 . 

21 . 

22 

23 

23. 


13 - 

13 

13 

12 . 

12. 

12. 

12 

12 

16 

16. 

16. 

17 

17 - 

18 

18. 

19 

20 

20. 

21. 


18. 

18 

18 

18 

17. 

1 7 • 

17 • 

17 

14 

14. 

15 

15 

15 - 

16 

16. 

17 

17 . 

18 

19 


23 • 

23 

23 

23 

22. 

22. 

22. 

22 

1 2. 

12. 

13 

13 . 

13. 

14 

14. 

15 

15 - 

16 

16. 


28. 

28. 

28 

28 

28 

27. 

27. 

27 • 

1°. 

11 

11 

11. 

11. 

12 

12. 

12. 

13 

13 - 

14 


33 - 

33 

33 

33 

33 

33 

32. 

32. 

jj 9 

9 

9 

9 - 

9 - 

10 

10. 

10. 

11 

11 . 

1 2 


38 . 

38 

38 

38 

38 

38 

38 

37 - 

7 

7 

7 - 

7 - 

8 

8 

8 

8. 

9 

9 

9 - 


43 • 

43 

43 

43 

43 

43 

43 

43 

5 . 

5 - 

5 - 

5 - 

6 

6 

6 

6. 

6 . 

7 

7 


48 

48 

48 

48 

48 

48 

48 

48 

a 3 - 

3 - 

3 - 

4 

4 

4 

4 

4 

4 • 

4 - 

4 • 


5 3 

53 

53 

53 

53 

53 

53 

53 

q 2 

2 

2 

2 

2 

2 

2 

2 

2 

2 . 

2. 



Ir8 

r8 

?8 

<8 

'58 

c 8 


0 

O 

O 

O 

O 

0 

O 

1 0 

O 

O 

0 


























































































































AZIMUTH . 


718£ 

Table LXII was computed with the mean place (declina¬ 
tion) of Polaris for each year. A closer result will be had by 
applying to the tabular results the correction from Table LXIII, 
which depends upon the difference of the mean and the apparent 
declinations of the star. 

Table LXIV will be used as follows: Find the hours of the 
time argument in the left-hand column of either page; then, 
between the heavy lines which inclose the hours, find the minutes 
in the column marked at the top with the current year. On 
the same horizontal line with the minutes the azimuth will be 
found under the given latitude, which is marked at the top of 
the right-hand half of each page. Thus, for 1904, time argument 
o h 43 111 , latitude 36°, find o h on left-hand page, and under 1904 
find 43“ on tenth line from the top, and on same line with the 
minutes, under latitude 36°, is the azimuth o° 17'. For 1908, 
time argument 9 11 33 , latitude 48°, the azimuth is i° ij', found 

on the twenty-first line from the top of right-hand page. 

If the exact time argument is not found in the table, the 
azimuth should be proportioned to the difference between the 
given and tabular values of said argument. 


PRIMARY AZIMUTHS . 


719 


313- Primary Azimuths .— 

EXAMPLE OF RECORD OF AZIMUTH OBSERVATION AT ANY 

POSITION OF STAR. 

(Station : West base, near Little Rock, Ark. Fauth 8", theod. No. 300. December 27, 1888. 

1 div. micr. = 2". 1 die. level = 3".) 


Object. 


Polaris 


E. base (mark). 
E. base (mark). 
Polaris. 


Polaris. 


E. base (mark). 
E. base (mark). 
Polaris . 


Time 

P.M. 

Level. 

Micrometer. 

Mean. 

Angle. 

West 

end. 

East 

end. 

A. 

B. 




Telescop 

>e direct. 




h. m. s. 

Div. 

Div. 

0 ' Div. 

0 7 Div. 

O / ft 



11 00 18 

I 3-9 

47 • 1 

346 00 14.8 

165 58 25.1 

345 59 39-9 




50.5 

10.2 





O / fP 








r 1 ! 5 3 2 30-0 


64.4 

57-3 







+ 7 -i 









101 32 18.1 

281 31 21.8 

101 32 09.9 






101 32 19.8 

281 31 19.7 

10I 32 00.q 


■ 115 34 16.1 

11 09 20 

5°-4 

10.3 

345 58 22.0 

165 57 01 4 

345 57 53-4 



13-8 

46.5 







64.2 

56.8 







+ 7 

•4 









Telescop 

e reverse. 




11 17 14 

50 5 

10.1 

211 28 29.0 

31 27 23.4 

211 28 22.4 

■ 



12 9 

46.6 







63-4 

56.7 





h ”5 35 53-8 


-f- 6.7 









327 05 06.7 

147 03 09.5 

327 04 16.2 






327 04 26.3 

147 03 00.6 

327 03 56.9 



11 26 22 

14.3 

46.3 

21I 27 IO.7 

31 26 07.4 

211 26 48.1 


115 37 °8.8 


50.1 

10.5 







64.4 

56.8 







+ 7 

.6 







SUMMARY OF RESULTS. 

(Station: West base, Arkansas. December 27, 1888.) 


Individual Results. 


First 
set... 


O III II 

294 '°& 2 (35.^0 


49 

34 


■ 42.3s R. 

21 k“ R - 

(37.650. 


4 1 

33 


Combined 

Results. 


allI 


\ 


38.80 


39-38 


Individual Results. 


O / // 


Second 
set.... 


29410 4 42 ; 4 4 } 43.90 D, 
40^8 |”33-6oR. 

45.0 [47-05 R- 

£.0 [33.150. 


Grand mean, 


Combined 

Results. 


O III 


38-75 


40.10 


294 10 39.26 

_ 













































































720 


AZIMUTH. 


314 . Reduction of Azimuth Observations. —The time of 
observation of a star is first to be corrected for the difference 
in longitude, assuming that standard time has been used, and 
for the error of the watch. It is then reduced from mean to 
sidereal time. From the sidereal time of observation is to be 
subtracted the right ascension of Polaris, if that star is used, . 
which is given in the Nautical Almanac, the result being the 
hour-angle or the sidereal time which has elapsed since it 
passed the meridian of the place of observation, given in 
hours, minutes, and seconds. This result is to be converted 
into degrees, minutes, and seconds. Then 

a sin t 

tan A — —-7- .... (140) 

1 — b cos t \ ^ J 

where a — sec 0 cot S ; 
tan 0 

A = angle between true north and the star. 

The angle between the star and the mark is to be corrected 
for level as follows: 


level corr. =- 


(w -f- zu') — (e -j- e f ) 


tan h. 


(141) 


where d — value of a division of the level; 
w _|_ w ' = readings of west end of level-bubble; 

= readings of east end of level-bubble; 
h — the angular elevation of pole-star. 






AZIMUTH AT EL ONGA T10N. 


721 


EXAMPLE OF REDUCTION. 


(Station : West base ; December 27, 1888. Observer, S. S. G.) 


Latitude — 34 45 26''.8. Longitude 92 0 13' 3i // .5 > 

Time of observation = T w . = n h oo m 18* 

Correction; ninetieth meridian time to g2°.2i5. =_ 3 54 

Watch slow; ninetieth meridian time. 4- 


Local mean time, T„ t . 

Correction; mean to sidereal time 
Right ascension mean sun. 

Sidereal time of observation. 

R. A. Polaris. 

Hour-angle, t . 


= 10 

51 

26 

rz: 

+1 

47 

18 

26 

36 

= 29 

19 

49 

— 1 

18 

25 

= 28 

01 

24 

- 24 




t (time) — 4 h oi m 24 s 
t (arc) = 60 0 2r / oo'' 


tan A — 


a sin t 

---where a — sec (p cot d, b — 

I — b cos t 


tan (p 
tan <5 


0 = 34° 45' 26".8 log sec = 0.0853539 

5 = 88 43 11.9 log cot = 8.3491690 


log tan = 9.8413076 
log tan =r 1.6508310 


log a ~ 8.4345229 log b 

log sint t 6o°2i'oo // =99390515 log cos / 


= 8.1904766 
= 9-6943423 


log a sin t 
log (1 — b cos t ) 


= 8.3735744 log — .0076704 =7.8848189 

= 9.9966559 -f 1.0000000 


log tan A 178° 38'08".o = 8.3769185 

angle to mark +115 32 30. o 

Level corr. —3. 8 = 


Az. of mark = 294 0 10' 34 r/ .2 =—- 

4 


0.9923296 =i-^cos^ 


— w -f w') -{e-\- e ')} tan h. 
Div. 


X 


7 -i 


X .694 = - 3". 8 


315. Azimuth at Elongation. —When observations for 
azimuth are to be made at elongation, it is necessary to know 
the mean time of elongation. This is computed by obtaining 
the hour-angle at elongation from the following equation : 


cos t e — tan 0 cot S .( r 4 2 ) 


The hour-angle plus the right ascension of the star gives 


























722 


AZIMUTH. 


the sidereal time of its western elongation, which, reduced to 
mean time, gives the local mean time in question. 

The azimuth of a pole-star at elongation is determined by 
the use of the equation 

sin A = sec 0 cos £. 043 ) 

4 

EXAMPLE OF COMPUTATION OF THE AZIMUTH AT ELONGA¬ 
TION, AND THE LOCAL MEAN TIMES OF BOTH ELONGATIONS 

OF POLARIS. 

(Latitude = <p — 40°. Meridian of Washington. November 28, 1891.) 

Sine, Azimuth at elongation — sec. (p cos < 5 . 
log sec 40° = 0.1157460 

log cos 8 88 ° 44' os".5 = 8.3439803 

log sine A 1 39 05. 8 = 8.4597263 

cos hour-angle at elongation, t e , =tan 0 cot <5 
log tan 40° = 9-9238i35 

log cot 8 88° 44' 05".5 = 8.3440862 

log cos t e 88 56 17. 5 = 8.2678997 

te = 5 h 55 m 45 s - 2 . 

Sidereal time western elongation, T s = R. A. Polaris -j- t e . 

R. A. Polaris.= I 1 * 19 1 ' 1 35.2 s 

te — 5 55 45.2 

Sidereal time western elongation, Ts— 7 15 20.4 
R. A. mean sun, a s .= 16 29 14.4 

Sidereal interval before noon, 1 .= 9 13 54.0 

Correction sidereal to mean interval = — 1 30.7 

Mean interval before noon. 9 12 23.3 Nov. 28. 

Local mean time, western elongation = 2 47 36.7 a.m., Nov. 28. 
Sidereal time E. elongation = 24"+a — t c — 19' 1 23™ 50.0 s 

a s — 16 29 14.4 

Sidereal interval after noon, /. ,....= 2 54 35.6 

Correction sidereal to mean interval.... = — o 28.6 

Local mean time eastern elongation.= 2 54 07.0 P.M., Nov. 28. 

Local mean time western elongation- = 2 47 36.7 A.M., Nov. 28. 

For longitudes west of Washington decrease times of elongation o*.66 
for each degree. 














CHAPTER XXXIV. 


LATITUDE. 

316. Methods of Determining Latitude.— 1. The most 
precise method known for determination of a terrestrial lati¬ 
tude is by measuring small differences of zenith distances of two 
stars with zenith telescope. (Art. 319.) 

2. The simplest method is by measuring the meridian 
zenith distance or altitude of a known star, though the result 
is relatively approximate only. It is only essential to follow 
a star near meridian until its altitude is greatest. The for¬ 
mula is 


z — z x R, 


and 


0 = 6 ± z, . . . . . . (144) 

sign of z depending on whether the star is north or south of 
the zenith. 

3. If the time be known , latitude may be determined by a 
single measured altitude of the sun or a star. (Art. 318.) This 
method gives fairly approximate results when time is known 
by a chronometer or watch to within two or three seconds, 
and is very useful in exploratory work. 

4. Time being known , latitude may be simply and quite 

723 


7 2 4 


LA TITUDE. 


accurately determined by measuring circummeridian altitudes 
of Polaris; this consists in applying the third method to 
Polaris. Then 

0 = h — p cos / -f- i / 2 s in i" • sin’ t . tan . (145) 

in which p — polar distance of Polaris or complement of 8 in 
seconds, which is about 5400". Tables for finding p and 
\p ' 1 sin 1" are given in the American Ephemeris. The best 
time of observation is when the star is at one of the culmina¬ 
tions. This method is especially adapted to the instruments 
available to the topographer, namely, a good theodolite or 
engineer’s transit and a good timepiece. 

5. Approximate latitude may be determined from an ob¬ 
servation on the sun at noon. (Art. 317.) This is a very useful 
method for the explorer or land surveyor. 

317. Approximate Solar Latitude —The following is a 
method of obtaining the approximate latitude from an obser¬ 
vation on the sun at noon: 

Measure two altitudes , one of the upper and the other of 
the lower limb of the sun, commencing before noon and watch¬ 
ing until the sun has reached its highest altitude. In order to 
eliminate errors of collimation, these two observations should 
be made on each limb with the telescope direct and inverted. 

Let r = refraction ; 

h — altitude of sun’s center; 

0 = latitude; 

d = sun’s declination at time of observation. 

The declination is taken from the Nautical Almanac for 
the date of observation, and increased or diminished by the 
hourly difference multiplied by the longitude from the locus 
of the almanac, expressed in hours. Then 


0 = 9°° ~ (/* - r - tf).(146} 


LATITUDE FROM AN OBSERVED ALTITUDE. 725 


EXAMPLE. 


April 16, 1898. Approximate longitude 136° 20', measured from map. 

Vertical circle reads, when pointing at sun’s upper limb. 36° 55' 

Vertical circle reads, when pointing at sun’s lower limb. 36 23 


Mean. 

Mean correction for index error 


36 ° 39 ' 
1' 


Apparent altitude. 36° 38' 

Refraction, always negative, enter table with arguments appar¬ 
ently negative. — 1' 

Altitude of sun’s center. .. 36° 37' 

Sun’s declination, April 16—Greenwich noon; from almanac .... — io° 14' 
Hourly change from almanac = 53", multiply by longitude and 
divide by 15: 


53 " X 136 
15 


—- 473 


// 


8 ' 


Altitude of celestial equator. 26° 15' 

Subtract from 90° gives latitude. 63° 45' 


A result to be relied upon within i' or 2supposing the vertical circle and 
collimation correct to within the same amount. 


318. Latitude from an Observed Altitude. —Latitude 
may be determined at sea or on an exploratory survey by 
measuring the altitude of a star or of the sun with a sextant, 
theodolite, or altazimuth. For this operation the time must 
be known, though the object observed may be in any position. 
The formula applicable is 

tan D = tan 6 sec t, .(14 7) 

cos (0 — D) = sin h sin D cosec d, . . . (148) 

in which d = declination of star; 

t — hour angle of star; 

D = auxiliary angle taken to simplify computation— 
it should be less than 90° and -f- or — accord¬ 
ing to algebraic sign of the tangent; 
h = altitude resulting from measurement after apply¬ 
ing all corrections. 
















726 


LA TITUDE . 


Although 0 — D may be positive or negative, the latitude 
of the place 0 is generally known with sufficient accuracy to 
decide this. 

The altitude h must be corrected for instrumental errors 
(Arts. 323 and 324), refraction (Arts. 322 and 325), and, in 
the case of the sun, for parallax and semi-diameter (Art. 301). 

319. Astronomic Transit and Zenith Telescope. —For 
the determinations of time and latitude separate transit instru¬ 
ments and zenith telescopes are sometimes employed. The 
astronomic transit is designed primarily for the determination 
of time when the telescope is in the plane of the meridian. 
Its essential parts are a telescope, an axis of revolution at right 
angles to the telescope, the supports for both, and a striding- 
level for the determination of the inclination of the axis. A 
zenith telescope is a somewhat differently constructed instru¬ 
ment provided with a large vertical circle and delicate level, 
and with a horizontal circle which turns with the upper part 
of the instrument much as does a theodolite. 

The most compact and useful instrument for determination 
of both latitude and time is a combination transit and zenith 
telescope , such as is used by the U. S. Geological Survey (Fig. 
181). This embodies the latest improvements in both instru¬ 
ments. It consists of a circular base resting upon three level¬ 
ing screws, and upon this base the whole instrument may re¬ 
volve when in use as a zenith telescope. About the base is a 
large graduated circle, provided with micrometer screw for 
slow motion to be used in setting the instrument and in ad¬ 
justing it in azimuth. The telescope of the above instrument 
has a focal distance of 27 inches, a clear aperture of 2.5 inches, 
and its magnifying power with diagonal eyepiece is 74 diam¬ 
eters. For use as a zenith telescope there is attached a verti¬ 
cal circle reading by vernier to 20", to which is fastened a 
delicate level. In the focus of the object-glass is a thread 
movable by means of a micrometer screw for the measurement 
of differences of zenith distances. 


AS 7 ROA 0 MIC TRANSIT AND ZENITH TELESCOPE. 727 



Fig- 181.—Astronomic Transit and Zenith Telescope 





























































































































































































































728 


LA TITUDE. 


For use as a transit the telescope is provided with a del¬ 
icate striding-level for measurement of inclination of the axis, 
and a reversing apparatus for turning the telescope in the 
wyes. The stationary reticule in the focus of the instrument 
consists of five threads for observing as many transits of the 
star. The reticule is illuminated by lamps, the light of which 
enters the hollow axis of the telescope and is reflected by 
a mirror into the eye. 

320. Latitude by Differences of Zenith Distances of 
Two Stars. —The zenith distance of a star on the meridian 
is the difference between the latitude of the station of ob¬ 
servation and the declination of the star; therefore the meas¬ 
urement of the meridional zenith distance of a known star 
furnishes a determination of the latitude. The most accurate 
method of determining the latitude of a place, and that gener¬ 
ally employed in geodetic operations, is that known as the 
Horrebow-Talcott method. In this, instead of the measure¬ 
ment of the absolute zenith distance of the star, the small 
difference of zenith distances of two stars culminating at 
about the same time on opposite sides of the zenith is meas¬ 
ured. Then 


z = 0 — <5 and 
z' — — 0, hence 


0 — ~ 2 ')- • • • (149) 

This method therefore requires that the difference (z — z') 
be measured. The stars must be so chosen that (z — z') may 
be measured by means of the micrometer in the telescope. A 
measurement of the latitude within 5" is possible by this 
method with a theodolite having full vertical circle. If now 
z' refer to the northern star, (z — z') in terms of the observed 


ERRORS AND PRECISION . 729 

micrometer readings becomes (M — M')r, in which r is the 
angular value of one turn of the micrometer screw. 

321. Errors and Precision of Latitude Determina¬ 
tions. —Latitude determinations by zenith telescope or transit- 
zenith telescope are subject to errors of three general kinds: 

1. External errors ; 

2. Instrumental errors; and 

3. Observer’s errors. 

No attempt will be made here to fully discuss these errors, 
as the kind of work which this volume is intended to explain 
is not of such high quality as either to warrant their correc¬ 
tion or a reduction of the results by least-square methods. 
These are fully explained in the more extended treatises 
already referred to. 

The external errors are those due to abnormal refraction, 
star places, and to defective declinations. The latter have 
probable errors sufficiently large to account for more than half 
of the error in the final result. The errors in the computed 
differential refractions are probably very small. Observer s 
errors are those made in bisecting a star and in reading the 
level and micrometer. They are of the kind known as per¬ 
sonal errors or those due to personal equation. Instrumental 
errors include those due to inclination of the micrometer line 
to the horizontal, to an erroneous level value, to inclination 
of the horizontal axis, to erroneous placing of the azimuth 
stops, to error of collimation, to an erroneous mean value 
of the micrometer screw, and to instability of relative posi¬ 
tions of different parts of the instrument. The errors of the 
first and second sources are small, but must be carefully 
guarded against. In the first instance the observer should 
study to make the bisectiondn the middle of the field. If 
the error from using an erroneous level is small, the level 
corrections will be small. 

In planning a series of observations the observer must de¬ 
termine the quality of the result desired which will fix for 


730 


LA TITUDE . 


him as to how many observations shall be made and how 
many separate pairs observed. Increasing either of these 
increases the cost of field- and office-work. The ratio of 
observation to pairs should be such as to give a maximum 
accuracy for a given expenditure. Extremes of practice are 
given by Hayford as 2 10 observations on 30 pairs each ob¬ 
served on seven nights, and 100 observations on 100 pairs each 
observed but once. The first is the practice of the U. S. Coast 
and Geodetic Survey. The practice of the U. S. Geological 
Survey is 100 observations on 20 pairs on each of five nights. 

322. Field-work of Observing Latitude.— The follow¬ 
ing description and example of the field-work of observing 
latitude is taken chiefly from Gannett’s “ Manual of the Top¬ 
ographic Methods of the U. S. Geological Survey T Before 
commencing the field-work a list of pairs of stars must be 
prepared, each pair of which shall have such zenith distances 
that they will culminate at nearly equal distance, one north 
and the other south of the zenith. Lists of such stars are 
published in the British Association Catalogue, various Green¬ 
wich catalogues, Safford’s Catalogue of the Wheeler Survey, 
and in various miscellaneous publications giving star lists pre¬ 
pared for special surveys. To prepare such a star list it is 
necessary to know approximately the latitude of the station 
and the right ascensions and declinations of the stars. When 
the declination of a star is known, the zenith distance is ob¬ 
tained by subtracting the latitude of the place from its declina¬ 
tion. The stars selected are such as culminate within a few 
minutes of one another and should be observed consecutively. 
In selecting them by pairs, therefore, only sufficient interval 
of time should be left between pairs to allow of the setting 
of the instrument. 

At the beginning of the observation the instrument should 
be placed in the line of the meridian and carefully collimated. 
At the approach to the meridian of the first star of the pair, 
the instrument should be set for it by the vertical circle, the 


FIELD-WORK OF OBSERVING LATITUDE . 731 

spirit-level upon that circle being made as nearly level as pos¬ 
sible. As the star traverses the field of the telescope, the 
movable thread in the reticule is kept upon it by means of a 
micrometer screw until it crosses the middle vertical thread, 
then the micrometer and the divisions of the level-bubble are 
read. Immediately, without disturbing the setting of the 
telescope, the entire instrument is revolved through 180° on 
its vertical axis, when it will point to the other side of the 
zenith at the same angle as before and will then be set for the 
opposite star. As this approaches culmination the same opera¬ 
tion is performed as before, reading the micrometer and the 
level again. 

For the determination of a latitude at least 20 such pairs 
of stars should be observed each evening, and the same pairs, 
if possible, should be observed upon several other evenings. 
The following example is taken from the observations at 
Rapid, South Dakota. 

EXAMPLE. —LIST OF STARS, FOR OBSERVATION WITH 

ZENITH TELESCOPE. 

(Station : Rapid, South Dakota. Approximate Latitude : 44 0 05 '.) 

(From Gannett’s Manual.) 


Name or 
Number. 
Safford’s 
Catalogue. 

Mag. 

Class. 

R. 

A. 

Dec. 

Zen. Dist. 

Setting. 




h. 

m. 

O 

/ 

O 

/ 


O 

/ 


7 Lacertae... 

4.0 

A A 

22 

27 

49 

43 

5 

38 

N. 

\ C 


N 

10 Lacertae.. 

5-o 

A A 

22 

34 

38 

29 

5 

36 

S. 

1 5 

37 


1539. 

6-5 

B 

22 

41 

45 

37 

1 

32 

N. 

t 1 

27 

N. 

1551. 

6-5 

A 

22 

47 

42 

42 

1 

23 

S. 

f 



1565 . 

6-5 

C 

22 

52 

38 

42 

5 

23 

s. 

i 5 

22 

S. 

1579. 

5-o 

A 

22 

59 

49 

26 

5 

21 

N. 

f 5 


1600. 

6.0 

A 

23 

08 

56 

34 

12 

29 

N. 

1,2 

I n 

N 

1633 . 

6-7 

B 

23 

18 

3i 

56 

12 

09 

S. 




1676 . 

5-6 

A 

23 

42 

67 

12 

23 

07 

N. 

[ 23 

05 

N. 

1686 . 

6-5 

A 

23 

47 

21 

03 

23 

02 

S. 

1702 . 

4 5 

A 

23 

52 

24 

32 

19 

33 

S. 

u 

31 

S. 

1722 . 

6-5 

B 

24 

00 

63 

35 

19 

30 

N. 

f 9 

j * 



































7 l 2 


LA TIT C/DE. 


EXAMPLE. —RECORD OF OBSERVATION. 

(Station: Rapid, South Dakota. Date: November 9, 1890. Instrument: Fauth combined 
transit and zenith telescope No. 534. Observer : S. S. G. Recorder: A. F. D.) 

(From Gannett’s Manual.) 


Star Name or 
Number. 

N. or 
S. 

Microm¬ 

eter 

Reading. 

Diff. 

Level. 

(N + S) 

— (N'+SO 

Remarks. 

N. 

S. 



Rev . 


Rev . 

Div . 

Div . 

Div . 


7 Lacertae... 

N. 

26.256 



39 9 

16. 7 

-f-56.6 


10 Lacertae.. 

S. 

24.052 

— 

2.204 

26. 5 

49-7 

— 76.2 









— 19.6 


1539 . 

N. 

30 432 



42.0 

18.7 

-I-60.7 


1551 . 

S. 

20.095 

— 

10-337 

21.9 

45 -o 

— 66 9 









— 6.2 


1565. 

S. 

25 164 



14.1 

37-6 

- 51-7 

Faint. 

1579 . 

N. 

26.703 

T 

1-539 

38.1 

15.0 

4 - 53-1 

Distinct. 








4- 1.4 


1600 . 

N. 

32.214 



37-5 

14.1 

4 " 5 1 • 6 


1633 . 

S. 

16.033 

— 

16.181 

19.9 

43 -i 

-63.0 

Faint. 








- II .4 


1676. 

N. 

26.656 



51.0 

28.0 

4 - 79-0 


1686 . 

S. 

17.684 

— 

8.972 

17.0 

39-6 

-56.6 









— 22.4 


1702 . 

S. 

25-345 



18.0 

40.9 

- 58.9 


1722. 

N. 

23.722 

+ 

1.623 

36.0 

132 

4-49.2 








• 

- 9-7 

• 


323. Determination of Level and Micrometer Constants. 

—Before proceeding with the reduction of latitude observa¬ 
tions, it is necessary to investigate the constants of the in¬ 
strument, to ascertain the value of a division of the latitude 
level, and of a division of the head of the micrometer screw. 

The value of a division of the head of the micrometer screw 
is measured by observing the transits of some close circum¬ 
polar star, when near elongation, across the movable thread; 
setting the thread repeatedly at regular intervals in advance 
of the star, and taking the time of its passage, with the read¬ 
ing of the micrometer. The precaution should be taken to 
read the latitude level occasionally and correct for it if neces¬ 
sary. This correction, which is to be applied to the observed 
time, is equal to one division of the level, in seconds of time, 





































MICROME 7 ER CONSTANTS. 


733 


divided by the cosine of the declination of the star and mul¬ 
tiplied by the level error, the average level reading being 
taken as the standard. 

The time from elongation of the star requires a correction 
in order to reduce the curve in which the star apparently 
travels to a vertical line. The hour-angle of the star is first 
obtained from the equation 

cos t 0 = cot <$ tan 0, .... (150) 

d being the star’s declination, and 0 the latitude. 

The chronometer time of elongation, 

^0 = a — t 0 — 6 t } .(151) 

in which a is the right ascension of the star obtained from the 
American Ephemeris, and $t the error of the chronometer. 

Having thus obtained the chronometric time of elongation, 
the correction in question is obtained from the observed inter¬ 
val of time of each observation before or after elongation, 
from tables in Appendix No. 14, U. S. Coast and Geodetic 
Survey Report for 1880, pp. 58 and 59, and in part in the 
following articles (Tables LXVII to LXX). A discussion of 
this subject will be found in the appendix above referred to, 
in Hayford’s Geodesy, pp. 174 to 181, and in Chauvenet’s 
Astronomy, vol. II. pp. 360 to 364. 

The times of observation thus corrected for level and 
distance from elongation, are then grouped in pairs, selected 
as being a certain number ot revolutions ot the micrometer 
apart, and the time intervals between the members of each 
pair obtained. The mean ot these, divided by the sum of 
revolutions which separate the members ot each pair, is yet to 
be corrected for differential refraction , which is derived from 
the following equation : 

R = 57".7 sin r sec' Z, . t , (I 52) 


734 LA titude. 



Mean level. 












































































































LEVEL CONSTANTS. 


735 


r being the value of a division of the micrometer, and Z the 
zenith distance of the star. Four-place logarithms (Tables V 
and VI) are sufficient for computing this correction, as it is 
small. On the preceding page is given an example of record 
and computation of the value of a revolution of the micrometer 
of combined instrument No. 534 of the Geological Survey. 

If d be the value of one division of the latitude level, and 
n and 5 the north and south readings; then if the numbering 
of the level-tube graduation increases each way from the 
middle, the inclination of the vertical axis i is 

* = 7 [(* + »')- e + - 0 ] • • • • (153) 

4 


The value of a division of the level is commonly measured 
with a level-trier. The latitude level may, however, be easily 
measured by means of the micrometer, the value of a revolu¬ 
tion of that being obtained by the following method : 

Point the telescope upon some well-defined terrestrial 
mark and set the level at an extreme reading near one end of 
the tube. Set the movable thread upon the object and read 
the micrometer and the level. Now move the telescope and 
level until the bubble is near the other end of the tube. 
Again set the movable thread upon the object and again read 
both micrometer and level. It is evident that the micrometer 
and the level have measured the same angle, and that the ratio 
between these readings equals that between a revolution of the 
micrometer and a level division. (See Example next page.) 

Secondly, when the star is observed off the line of collima- 
tion, the instrument remaining in the plane of the meridian, 
then 


m 


2 sin . 2 sin 

—7- rr sin 6 cos 6 , or m ——:-77-• 4 sin 20, 

sin I sin 1 


(■54) 


and the correction to the latitude is half of this quantity, 




7 36 


LA T1TUDE. 


EXAMPLE.—DETERMINATION OF VALUE OF ONE DIVISION OF 

LATITUDE LEVEL No. 534 . 


(By comparison with micrometer screw 534.) 

(From Gannett’s Manual.) 



Level. 

Difference. 



Micrometer. 





aa. 

ab. 

N. 

S. 

Microm. 

Level. 




r. 

8.025 

8.508 

d. 

47-3 

20.7 

d. 

29.2 

02.7 

b. 

d. 

48.3 

a . 

d. 

26.55 

704.9 

1283 

8.509 

7.984 

18.9 

49.8 

OI .O 

31.0 

52.5 

30-45 

927.2 

1599 

8.511 

8.045 

18.5 

47.2 

00.6 
29.1 

46.6 

28.60 

818.O 

1333 

9.076 

8.604 

18.7 

46.0 

00.8 

28.0 

47.2 

27.25 

742.6 

1286 

9.442 

9.009 

23-7 

48.0 

06.0 

30.0 

43-3 

24.15 

583.2 

IO46 

10.055 

9-574 

21.8 

48.0 

04.0 
30.1 

48.1 

26.15 

683.8 

1258 

10.661 

10.212 

24.0 

50.7 

06.1 

33-0 

44.9 

26.80 

7 l 8.2 

1203 

11.771 
11.252 

18.3 

48.3 

00.7 

3 i -9 

5 i -9 

30.60 

936.4 

1588 

12.328 

11.872 

20.0 
46.1 

02.3 

28.5 

45-6 

26.15 

683.8 

II92 

12.869 

12.438 

22.2 

47-7 

04.6 

30.0 

43 -i 

25-45 

647.7 

IO97 

13.468 

13.080 

23.0 

44-5 

05-3 

26.9 

38.8 

21-55 

464.4 

836 

14.146 
13.702 

20.1 

45-4 

02.4 

27.8 

44.4 

25-35 

642.6 

1125 

14-758 

14.282 

22.3 

48.6 

04.8 

31-0 

47-6 

26.25 

689. I 

1249 

Sum.. 






16095 






9 Z 4 J -9 


l °S . 16095. = 4.20669 

a- c> . 9241.9 = 6.03424 

log 1 div. micrometer. = 0.87066 


l °S . 16095. = 4.20669 

a- c> . 9241.9 = 6.03424 


1 div. level 


= i".320 log = 0.12059 
































LEVEL AND MICROMETER CONSTANTS. 737 

whether the star be north or south; and if the two stars form¬ 
ing a pair are observed off the line of collimation, two such 
corrections, separately computed, must be added to the lati¬ 
tude. If the stars should be south of the equator, the essen¬ 
tial sign of the correction is negative. The value of m for 
every 5° of declination is given in the following table: 


Table LXVI. 

VALUES OF rn FOR EVERY 5 ° 8 . 



IOJ. 

i 5 J- 

VOS. 

25*. 

30 s. 

35s- 

40J. 

45 f * 

50^. 

55 ** 

60s. 


6 

n 

n 

// 

// 

n 

n 

tt 

n 

n 

// 

ft 

6 

5 ° 

.00 

.01 

.02 

•03 

.04 

.06 

.08 

. 10 

.12 

.14 

•17 

85 ° 

10 

.01 

.02 

.04 

.06 

.08 

. 11 

• 15 

.19 

• 23 

.28 

•34 

80 

15 

.01 

•03 

•05 

• OQ 

. 12 

• 17 

.22 

.28 

• 34 

.41 

•49 

75 

20 

.02 

.04 

.07 

. 11 

. 16 

. 22 

.28 

• 36 

• 44 

•53 

•63 

70 

25 

.02 

.05 

.08 

•13 

.19 

. 26 

• 34 

.42 

• 52 

•63 

•75 

65 

30 

.02 

•05 

.09 

•15 

.21 

.29 

• 38 

.48 

.59 

• 7 i 

.85 

60 

35 

• 03 

.06 

. IO 

. l6 

.23 

• 31 

.41 

• 53 

.64 

• 77 

•92 

55 

40 

•03 

.06 

. II 

•17 

.24 

• 33 

• 43 

.54 

.67 

.81 

•97 

50 

45 

• 03 

.06 

. II 

•17 

• 25 

• 33 

• 44 

.55 

.68 

. 82 

.98 

45 


Reduction of Observations on Close Circumpolar Stars, 
Made in Determining the Value of a Revolution of the Microm¬ 
eter. —Let t — difference of time of observation and elonga¬ 
tion of the star expressed in seconds, and z" — number of 
seconds of arc in the direction of the vertical from elonga¬ 
tion, then 

„ cos 6 sin t 


for which we can write 

*" = 15 costfj* — |(i 5 sin i")V 8 }. . . (155) 

It is convenient to apply the term £(15 sin T')V S to the 
observed time of noting either elongation, additive to the 


































738 


LA TITUDE. 


observed time before, and subtractive after. The following 
table gives the value of -^(15 sin CfC, a ^ s0 °f ^ ie additional 
term — T ^- 0 .( 15 sin i")Y 5 when sensible, for every minute of 
time from elongation to 65’”. 

Table LXVII. 

REDUCTION OF OBSERVATIONS ON CLOSE CIRCUMPOLAR 

STARS. 


(From Appendix 14, U. S Coast and Geodetic Survey Report for 1880.) 


t 

Term. 

t 

Term. 


t 

Term. 

/ 

Term. 

t 

Term. 


t 

Term. 

m . 

s. 

m . 

s. 


in. 


m. 

s. 

IK . 

s. 


m. 

s. 

6 

0.0 

16 

0.8 


26 

3-3 

36 

8.9 

46 

• 18.5 


56 

33-3 

7 

0.1 

17 

0.9 


27 

3-7 

37 

9.6 

47 

19.7 


57 

35-1 

8 

0.1 

18 

I. I 


28 

4.2 

33 

10.4 

48 

21 .O 


58 

37 -o 

9 

0.1 

19 

1-3 


29 

4.6 

39 

n -3 

49 

22.3 


59 

39 *o 

10 

0.2 

20 

1-5 


30 

5-1 

40 

12.2 

50 

23-7 


60 

41.0 

11 

0.2 

21 

1.8 


31 

5-7 

4 i 

i 3 -i 

5 i 

25.2 


61 

43-1 

12 

o -3 

22 

2.0 


32 

6 2 

42 

14.1 

52 

26.7 


62 

45-2 

13 

0.4 

23 

2-3 


33 

6.8 

43 

15 -1 

53 

28.3 


63 

47-4 

14 

0.5 

24 

2.6 


34 

7-5 

44 

16.2 

54 

29.9 


64 

49-7 

15 

0.6 

25 

3 -o 


35 

8.2 

45 

17-3 

55 

31.6 


65 

52 .1 


324. Corrections to Observations for Latitude by 
Talcotts Method. — Correction for Differential Refractio7i. 
—The difference of refraction for any pair of stars is so 
small that we can neglect the variation in the state of the 
atmosphere at the time of the observation from that mean 
state supposed in the refraction tables. The refraction 
fbeing nearly proportional to the tangent of the zenith dis¬ 
tance, the difference of refraction for the two stars will be 
given by 

R — R — 57".7 sin (z — z') sec 3 z\ . . (156) 

and since the difference of zenith distances is measured by 
the micrometer, the following table of correction to the lati- 
































LATITUDES BY TALCOTT'S METHOD . 739 

tude for differential refraction has been prepared for the argu¬ 
ment J difference of zenith distance, or J difference of microm¬ 
eter reading, on the side, and the argument “ zenith distance ” 
on the top. The sign of the correction is the same as that of 
the micrometer difference. 

Table LXVIII. 


CORRECTION FOR DIFFERENTIAL REFRACTION. 

(From Appendix 14, U. S. Coast and Geodetic Survey Report for 1880.) 


$ Diff. in 
Zenith 
Distance. 

Zenith Distance. 

o° 

IO° 

20° 

25° 

30° 

35° 

t 

ff 

tf 

If 

t • 

// 

// 

O 

.00 

.OO 

.OO 

.OO 

.OO 

.OO 

0-5 

.01 

• OI 

.OI 

.OI 

.01 

• OI 

I 

.02 

.02 

.02 

.02 

.02 

.02 

1-5 

.02 

•03 

.03 

•03 

•03 

•03 

2 

•03 

•03 

.04 

.04 

.04 

•05 

2-5 

.04 

.04 

•05 

•05 

.05 

.06 

3 

•05 

•05 

.06 

.06 

.07 

.08 

3-5 

.06 

.06 

.07 

.07 

.08 

.09 

4 

.07 

•07 

.08 

.08 

.09 

. IO 

4-5 

.08 

.08 

.09 

.09 

. IO 

.11 

5 

.08 

.09 

. IO 

. IO 

. II 

•13 

5-5 

.09 

. IO 

. IO 

. II 

. 12 

.14 

6 

.IO 

. IO 

. II 

. 12 

.13 

•15 

6.5 

.11 

. II 

. 12 

•13 

.14 

. l6 

7 

. 12 

. 12 

•13 

•14 

•15 

.18 

7-5 

.13 

•13 

.14 

•15 

. l6 

.19 

8 

•13 

.14 

.15 

. l6 

. l8 

.21 

8.5 

.14 

•15 

. l6 

•17 

.19 

. 22 

9 

•15 

. l6 

•17 

. l8 

.20 

•23 

9-5 

. l6 

.17 

. l8 

.20 

.21 

. 24 

10 

•17 

. l8 

.19 

.21 

•23 

. 26 

10.5 

.18 

.19 

.20 

.22 

.24 

•27 

11 

. l8 

.19 

.21 

•23 

•25 

. 28 

11 • 5 

.19 

. 20 

.22 

.24 

. 26 

•30 

12 

. 20 

. 21 

•23 

•25 

.27 

•31 


Reduction to the Meridian .—First, when the line of colli- 
mation of the telescope is off the meridian, the instrument 



















740 


LA TITUDE. 


having been revolved in azimuth and the star observed at the 
hour-angle r, near the middle thread, then 

2 sin 2 \t cos 0 cos £ 

j} • ; » , • • • ( I 5 7 ) 

sin i sin C v J / 

and the correction to the latitude, if the two stars are ob¬ 
served off the meridian, is 


The value of 


Cor. 0 = \(m! — in). 
2 sin 3 


( 158 ) 


sin i 


// 


for every second of time up to two 


minutes (a star being rarely observed at a -greater distance 
than this from the meridian in zenith-telescope observations) 
is given in the following table: 

Table LXIX. 

2 sin 2 \r 


VALUES OF 


sin i 


T 

Term. 

T 

Term. 

T 

Term. 

T 

Term. 

1 

1 T 

Term. 

T 

Term. 

J. 

// 


// 

s. 

// 

S. 

// 

s. 

// 

S. 

// 

I 

0.00 

21 

O.24 

4 i 

O.91 

61 

2.03 

81 

3.58 

101 

5-56 

2 

0.00 

22 

0.26 

42 

0.96 

62 

2 . IO 

82 

3-67 

102 

5-67 

3 

0.00 

23 

0.28 

43 

I .OI 

63 

2 . l6 

83 

3-76 

103 

5-78 

4 

0.01 

24 

0.3I 

44 

I .06 

64 

2.23 

24 

3-85 

104 

5-90 

5 

0.01 

25 

0-34 

45 

I. IO 

65 

2.31 

85 

3-94 

105 

6.01 

6 

0.02 

26 

0-37 

46 

I • 15 

66 

2.33 

86 

4-03 

106 

6.13 

7 

0.02 

27 

0.40 

47 

1 .20 

67 

2-45 

87 

4.12 

107 

6.24 

8 

0.03 

28 

0-43 

48 

I . 26 

68 

' 2.52 

88 

4.22 

108 

6.36 

9 

0.04 

29 

0.46 

49 

1 • 3 1 

69 

2.60 

89 

4-32 

109 

6.48 

10 

0.05 

30 

O.49 

50 

1.36 

70 

2.67 

90 

4.42 

110 

6.60 

ii 

0.06 

3 i 

O.52 

5 i 

1.42 

7 i 

2-75 

9 i 

4-52 

hi 

6.72 

12 

0.08 

32 

O. 56 

52 

1.48 

72 

2.83 

92 

4.62 

112 

6.84 

13 

0.09 

33 

0-59 

53 

i -53 

73 

2.91 

93 

4.72 

H 3 

6.96 

14 

0. II 

34 

0.63 

54 

i -59 

74 

2.99 

94 

4.82 

114 

7.09 

15 

0.12 

35 

O. 67 

55 

1.65 

75 

3*07 

95 

4.92 

1 15 

7.21 

16 

0.14 

36 

O.71 

56 

1.71 

76 

3 -i 5 

96 

5-03 

116 

7’34 

17 

0.16 

37 

o -75 

57 

i -77 

77 

3-23 

97 

5-13 

117 

7.46 

18 

0.18 

33 

0.80 

58 

1.83 

78 

3-32 

98 

5-24 

118 

7.60 

19 

0. 20 

39 

0.83 

59 

1.89 

79 

3-40 

99 

5-34 

119 

7.72 

20 

0.22 

40 

0.87 

60 

1.96 

80 

3-49 

100 

5-45 

120 

7.85 










































APPARENT DECLINATIONS OF STARS. 741 

The Determination of Apparent Declinations of Stars Used 
is the next step. Whenever possible these should be taken 
from the American Ephemeris, the Berliner Jahrbuch, or 
other reliable sources. The positions of stars are also given 
in Safford’s Catalogue for the epoch of 1875, together with 
the annual precession and proper motion. The declinations 
given there should be revised by the aid of more recent cat¬ 
alogues, particularly with reference to stars of the class C. 
The annual precession and proper motion multiplied by the 
number of years which have elapsed, applied with the effect 
of secular variation in precession, give the declination at the 
beginning of the year. To reduce from mean place at the 
beginning of the year to apparent place at any date may, with 
the aid of the Ephemeris, be put in the following form for 
apparent right ascension and declination at a stated time : 

« = Ct, +/ + t/x + T \g- sin (G + a,) tan d, 

-f- j-jk sin ( 7 / -f- a 0 ) sec d 0 _in time; . (159) 

S = 6 ° + tja' -\- g cos (G + a 0 ) -f- h cos (H 4 - cr 0 ) sin d 0 

+ i cos d 0 _in arc;. 06 °) 

in which a Q and d 0 = right ascension and declination at be¬ 
ginning of year; 

r = elapsed portion of fictitious year ex¬ 
pressed in units of one year as given 
in the Ephemeris; 

ju and ju' = annual proper motions in right ascen¬ 
sion and declination ; 

H, G, /, g, h , and i = quantities called independent star num¬ 
bers and are given in the Ephemeris. 


742 


LA TITUDE . 


EXAMPLE.—COMPUTATION OF APPARENT DECLINATION OF 
STAR 1539 (SAFFORD’S CATALOGUE) 

AT ITS TRANSIT AT RAPID, SOUTH DAKOTA, NOV. 9, 189O. 

(S. S. Gannett, Computer.) 

< 5 0 = 45 0 38' n".86 ; a 0 = 340° 23' = 22 h 4i m 32 s . 


Rapid is west of Washington. i h 45 m 

Rapid sidereal time of transit or a. 22 41^ 

Washington sidereal time. 24 264 

Sidereal time of mean midnight, Nov. 9 (Am. Ephemeris, p. 383). 27 17 
Hence sidereal interval before Washington midnight for stated 

time is. 2 s°h 

Equivalent to.:. 0.12 day 

The Ephemeris, p. 291, gives the following values Nov. 8, Washington 
mean midnight: 

t G H log g log h log i 


.86 316° 43' 40° 45' +1.0096 +1.2938. +0.7460 

Nov. 9 

.86 346 56 39 46 + 1.0103 +1.2945 +0.7377 

By interpolating the values at time of observation at Rapid .12 day 
before Washington midnight, Nov. 9, 1890, in accordance with formula 
(160), the computation of S is : 


G 

346 ° 55 ' 

H 

39 ° 53 ' 

log g log h log * 

1.0102 I.2944 O.7387 

lo gg 

log cos ( G + a 0 ) 327 0 18' 

= 1.0102 

= 9 - 925 I 

£0 = 45° 38' J 1".85 
r/P = .86 X — o".03 = — 0.03 

log g COS (G + a 0 ) 
g cos ( G + a 0 ) 

= 0.9353 

= 8". 62 

= + 8.62 

log h 

log cos (//’ + do) 20° 16' 
log sin S 0 

= 1.2944 

= 9.9722 

= 9-8543 



1.1209 


h cos (AT+ a 0 ) sin S 0 

= I 3 "- 2 I 

= + 13.21 

log i 
log cos < 5 0 

0.7387 

9.8446 



4 - 3.83 


0.5833 

i cos S = 3".83 


Apparent declination 
at time of observation 
Nov. 9, 1890. 


45° 38' 37"-48 















REDUCTION OF LATITUDE OBSERVATIONS. 743 

325. Reduction of Latitude Observations _With all this 

preliminary work done, the final reduction of latitude obser¬ 
vations is a comparatively simple matter. Grouping the 
observations by pairs, the mean decimation of each pair is 
obtained, the corrections for difference of micrometer readings 

o 

and levels are applied, with a small correction for differential 
refraction, and the result is the desired latitude. 

Applying the foregoing corrections to formula (149), we 
have the following working formula for reduction of latitude 
observations: 

<p = + S') + (M - M') r ~ + 7[(» + n') - (s + /)] 

z 4 

+ i (R-R') + j-f ( 169 ) 


EXAMPLE.—REDUCTION OF LATITUDE OBSERVATIONS. 

(Station : Rapid, South Dakota. November 9, 1890. Half rev. micrometer = 37".900, 

One div. level = i".33.) 


Date. 

Star Numbers. 



5(Sj 4 S 2 ) 




0 / tt 

of tf 

Of tf 

Nov. 9... . 

( 7 Lacert and ) 

1 10 Lacert. f 

49 42 87.33 

38 29 04.60 

44 06 15.97 


1539 

1551 

45 38 37-48 

42 44 04.63 

11 21.06 


1565 

1579 

38 43 39-78 

49 27 41.04 

05 40.41 


1600 

1633 

56 34 06.66 

3 i 55 56.91 

15 01.78 


1676 

1686 

67 12 10.93 

21 03 54.02 

08 02.48 


1702 

1722 

24 32 09.04 

63 35 27.34 

03 48.19 


Star Numbers. 

Corrections. 

Latitude 
n . 

Weight 

P - 

l 

p . n. 

Microm. 

Level. 

Refr. 

{ 7 Lacert and ) 

\ 10 Lacert. f 

1539 1551 

t 565 1579 

1600 1633 

1676 1686 

1702 1722 

t if 

- I 23.53 

- 6 31.77 

- 0 58.33 

- 10 13.25 

- 3 08.43 
4- 1 01.51 

// 

- 6.51 

— 2.06 
-j— O • 46 

- 3-78 

- 7-44 

- 3-22 

ff 

-•03 

— . II 

-•03 

-.19 

--07 
-|- .02 

Of ft 

44 04 45.90 

47.12 

42.51 

44-56 

46.54 

46.50 

.98 

.90 

•79 

.90 

-93 

.90 

ft 

5.78 

6.4I 
I.98 
4. IO 
6.08 
5-85 

5-40 

30.20 


NovemDcr 9. Weighted mean = 44’ 04' 4 S "-S 9 • 










































CHAPTER XXXV. 


LONGITUDE. 

326. Determination of Longitude. —Determining the lon¬ 
gitude of a point on the surface of the earth consists in find¬ 
ing the angle between the two meridian planes passing 
through the station and a reference meridian. In the United 
States, Greenwich, England, is generally accepted as the zero 
of longitude. Time and arc are interchangeable (Art. 304), 
differences in longitude may be expressed in time or angle. 
Thus 24 hours equals 360°, 1 hour equals 15 °, 1 minute of time 
equals 15' and 1 second equals 15" of arc (Tables LVI to LIX). 
Therefore the angle between the two meridian planes above 
described is the same as the differences of the local times of 
the two stations. Accordingly, to determine the longitude of 
a station is to determine the differences between the local time 
at Greenwich and the local time of that station (Art. 305), 
generally referred to some nearer station the longitude of 
which is already known. 

327. Astronomic Positions : Cost, Speed, and Accu¬ 
racy. —Practically the whole expense involved in determining 
the latitude, longitude, and azimuth of a station is included in 
the telegraphic exchange of signals and time observations for 
longitude, the additional observations required to determine 
latitude and azimuth being made in the meanwhile. 

The U. S. Coast and Geodetic Survey determines longi¬ 
tudes of prime importance at an average cost of $1500 per 
station. The observations are made by transit instrument for 
time and telegraphic exchange of clock signals on five nights. 

744 


LONGITUDE BY CHRONOMETERS. 


745 


The observers then change stations and repeat the same obser¬ 
vation on five additional nights, making a total of ten nights, 
requiring about six weeks of actual time. The probable error 
of a location is ± o.oi second of time, equivalent to from io 
to 15 feet in distance. 

The U. S. Geological Survey determines longitudes at a 
cost of about $500 per station. The method is by exchange 
of telegraphic signals, as in the Coast Survey, but on four 
nights only, the personal equation being determined on four 
other nights either preceding or following the field season. 
Accordingly, a determination of personal equation by the 
Geological Survey method serves for from three to four longi¬ 
tude determinations in a season, the average time per station, 
including observations for personal equation, being ten days to 
two weeks. The probable error of such a determination is ± .03 
seconds of time, equivalent to from 30 to 45 feet in distance. 

328. Longitude by Chronometers. —When it is imprac¬ 
ticable to determine longitude by telegraphic exchange of 
signals (Art. 330), the same principle may be employed be¬ 
tween two intervisible stations, as points on a shore line or 
the summits of mountain peaks, by flashes of light. 

The simpler and more usual way, however, of determining- 
longitudes in the absence of the telegraph is by means of 
chronometers or chronometer watches carried from some 
point the longitude of which is known to that at which it is 
to be determined. This performs the same purpose as the 
telegraph by comparing local times at the two stations. 

The mode of determining longitudes with chronometers is 
to observe transits of stars on as many nights as practicable, gen¬ 
erally from 10 to 50, catching the transit by eye and the chro¬ 
nometer beat by ear. At the known station there should be 
from 2 to 4 chronometers, part set to sidereal and part to mean 
time, and these should remain stationary and protected from 
changes of temperature. At the new station there should be 
a similar number of stationary instruments. Finally, several 


LONGITUDE . 


746 

chronometers, part set to mean and part set to sidereal time, 
should be carried back and forth between the two stations. 

The method of observing is to compare the moving with the 
stationary chronometers, and these compared with the transit 
observations serve to determine the error of each chronometer. 
The moving chronometer must be handled with the greatest 
possible care, and the results cannot be satisfactory where 
they are carried on wagons, or on the backs of animals. They 
may be carried with fairly satisfactory results in the hand, 
however. Where the mode of travel is rough, chronometer 
watches will give as satisfactory results as can be attained by 
attempting to transport large chronometers. 

The object of having chronometers set to both sidereal and 
mean time is similar to that of reading a vernier. The side¬ 
real chronometer gains gradually on the mean-time chronom¬ 
eter, and about once in three minutes the two chronometers 
tick exactly together. 

The mode of computing chronometric longitudes consists in 
applying to the time of a mean-time chronometer the correc¬ 
tion to local mean time, the result being local mean solar time. 
This must then be reduced to sidereal interval to give sidereal 
interval from preceding mean noon. The time of sidereal pre¬ 
ceding mean noon must then be applied, giving local sidereal 
time. This compared with the time of the sidereal chronom¬ 
eter gives the correction to the latter. 

329. Longitude by Lunar Distances. —If the direct 
methods of determining longitude are unavailable, such as 
those by telegraphic exchange of time signals with the chron¬ 
ograph (Art. 330) or by means of chronometers (Art. 328), 
there remains but one other method of determining longitude, 
dependent upon the motion of the moon. The position of the 
moon has been determined frequently at fixed observatories. 
As a result its orbit and its various perturbations have been 
computed. Tables giving the right ascension and declination 
of the moon for every hour, and other tables defining its place, 


LONGITUDE BY LUNAR DISTANCES. 


747 


are to be found in the American Ephemeris. If the topog- 
rapher wishes to determine longitude by the moon, he deter¬ 
mines its position and notes the local time at which his obser¬ 
vation was made. Then by consulting the Ephemeris and 
finding what interval by Greenwich time the moon was actu¬ 
ally in the position in which he observed it, the difference 
between this time and the local time of his observation is 
longitude reckoned from Greenwich. 

The various methods by which the position of the moon 
may be determined are all approximate, and the field-work 
connected with the making of these observations is laborious 
considering the inferior quality of the results. The attain¬ 
ment of accuracy by any method involving the moon is diffi¬ 
cult, because the moon requires about 27J days to make one 
complete circuit in its orbit about the earth. The apparent 
motion of the moon among the stars is accordingly as fast 
as the apparent motion of the stars relative to the observer’s 
meridian, which furnishes his measure of time. Any error 
in determining the position of the moon is accordingly multi¬ 
plied by at least 27 when converted into time. Moreover, the 
motion of the moon is so difficult to compute that its posi¬ 
tions at various times as given in the Ephemeris are in error 
by amounts which become whole seconds when multiplied 
by 27. Finally, the limb or edge of the visible disk of the 
moon, which is the object really observed, is seen as a ragged 
outline which makes it difficult to use for purposes of measure¬ 
ment. The computations required for the determination of 
longitude by lunar observations are long and complicated, and 
the theories involved require much study for their mastery. 
Accordingly, no attempt will be made to explain here the 
methods of determining longitude by lunar observations, 
reference being made to Doolittle’s Practical Astronomy, 
the American Ephemeris, and to Chauvenet’s Astronomy. 

Recently there has been devised by Captain E. H. Hills of 
the British Army a method of determining longitude by photo- 


748 


LONGITUDE. 


graphs of the moon and of one or more bright stars of ap¬ 
proximately the same declination. In 1895 Captain Hills 
took advantage of the despatch of a surveying expedition to 
the Niger River in Africa to carry out a series of field experi¬ 
ments for the determination of photographic longitudes. The 
results obtained are reported as most encouraging. No diffi¬ 
culty was experienced in the field although the observer was 
quite new to the work. The only apparent disadvantage of 
this method, and one which is not serious, rests on the fact 
that the results are not at once available, although this is 
generally of small moment, as the office measurement and 
computation can be done at leisure and with access to ac¬ 
curate measuring micrometers. Underlying the principle of 
this method is that of obtaining a photograph of the moon 
and the traces of one or more bright stars. The position of 
the moon is determined at the time of taking the photograph 
by means of some angle-reading instrument attached to or 
separate from the camera. After the exposure has been made 
on the moon the time is noted which elapses between it and 
the passage of some bright star across the field of the camera 
as denoted by the cross-hairs of the finding instrument. 
When an exposure of some duration is made on a star so that 
it shall leave a trace on the plate, or, in fact, several expo¬ 
sures are made as explained hereafter, the declination of the 
moon and stars being known and the time which has elapsed, 
these quantities, with the micrometric measurement of the 
distance between the limb of the moon and the star trace, 
give all the data from which to compute the longitude. 

In Chapter XXXVII are given a description and an exam¬ 
ple of the method of determining a photographic longitude, 
prepared by Mr. Wm. J. Peters of the U. S. Geological Survey. 

330. Longitude by Chronograph. —As already explained, 
all methods of determining longitude are reduced to deter¬ 
mining the differences of local times and converting these into 
differences in longitude (Art. 305). The most accurate method 




LONGITUDE BY CHRONOGRAPH . 749 

of determining time differences is by meridian transit observa¬ 
tions (Art. 308) for time at two stations, and the comparison 
of the results by the exchange of telegraphic signals. The 
operation consists in the observation of stars for time with a 
transit instrument of the type described in Article 319. 
These stars are observed in sets by previous agreement be¬ 
tween the observers at the station the longitude of which is 
known and that at which it is to be determined. At some 
time about the middle of the night’s observations, between 
two sets of time observations, arbitrary signals are exchanged 
by telegraph between the two stations, and these serve to com¬ 
pare the chronometers and thus to compare the local times at 
the two places as determined during the star observations. 

For the purpose of recording the time of transit of stars 
as observed with the transit an instrument called a chrono¬ 
graph is used. This consists of a drum upon which is wound 
a strip of paper kept in revolution by clockwork controlled 
by an escapement (Fig. 182). A pen carried by a car which 
travels slowly in a direction parallel to the axis of the cylin¬ 
der traces a line on the drum. This pen is held in place by a 
magnet carried also upon the car, and as long as the current 
from the battery passes through the coil and thus holds the 
armature the pen traces an unbroken spiral line. If the cur¬ 
rent is suddenly broken or destroyed, as by a touch of the ob¬ 
serving key, the armature is freed in an instant and a jog is 
made in the line. The batteries employed with this apparatus 
are the ordinary zinc, copper, and sulphate of copper appara¬ 
tuses of four cells. Dry batteries are also used successfully. 

As a part of this apparatus a break-circuit chronometer is 
used which differs from ordinary chronometers in that it is 
arranged to break an electric circuit temporarily at regular 
intervals. Those used in the U. S. Geological Survey break 
circuit every two seconds, the end of a minute being indicated 
by a break at the 59th as well as 60th second. One of these 
chronometers being connected with a battery and the chrono- 


750 


LONGITUDE. 



Fig. 182.—Chronograph 



























































































































































































































OBSERVING FOB TIME. 


75 e 


graph being introduced in the same circuit, the beginning of 
every second is recorded upon the chronograph battery by a 
jog, and the distance between any two jogs represents there¬ 
fore 2 seconds. The observer at the transit watches a star 
near the meridian, and as it crosses a thread in the telescope 
he presses an observing-key which is in circuit with the chron¬ 
ograph, and thus records by a jog on the chronograph sheet 
the time of passage between the threads. 

331. Observing for Time. —The transit being mounted,, 
leveled, and adjusted in the meridian as described in Article 
322, and the chronograph set up and running connected in a 
circuit with a battery, a chronometer, and a telegraph key, 
time observations are made in the following manner: 

A list of time stars should be consulted, as that given in 
the Berliner Jahrbuch, this being one of the fullest lists which 
give day places. Stars are selected north and south of the 
zenith so that the azimuth errors will balance one another as 
nearly as possible. On the approach of the selected star to 
the meridian the telescope is set by means of the vertical 
circle for the altitude of the star above the horizon, as 
determined from the declination and latitude. As the star 
crosses each thread in the reticule the fact is recorded upon 
the chronograph sheet by the observer pressing the observing 
key. At least four time stars, as those between the equator 
and zenith, are designated, and one circumpolar star should be 
observed and the telescope be reversed in the wyes and a 
similar set be observed. Two such half-sets with the reversal: 
of the telescope between gives an accurate determination of. 
time. The same sets of stars are by previous agreement 
observed at each station. 

Between observations upon any two stars the striding- 
level should be placed upon the pivots of the instrument and 
readings taken to ascertain the departure of the axis from a 
horizontal position. In order to avoid unequal expansion of 
the pivots from unequal heating, both bull’s-eye lamps must 


752 


LONG ITUDE. 


be lighted and placed in their stands, in order that both pivots 
may be equally heated. After the comparison of chronom¬ 
eters at the two stations, to be hereafter described, a similar set 
of stars should be observed, thus giving rate of the chronometer. 

332. Reduction of Time Observations. —Certain con¬ 
stants of the transits should be measured before proceeding 
with the reduction of time observations. The value of a division 
of the striding-level should be measured by means of a level- 
trier. The equatorial interval of time between each of the 
threads and the mean of all the threads should be obtained, 
as it is not infrequently needed in utilizing broken or imper¬ 
fect observations. These can best be obtained from observa¬ 
tions on slow-moving stars, but any stars may be used for the 
purpose. The intervals as observed are reduced to the equa¬ 
tor by multiplying them by the cosine of the declination of 
the star observed. The object of these observations is specifi¬ 
cally the determination of the error of the chronometer . 
This error equals the right ascension of a star minus its 
observed time of transit, corrected for certain instrumental 
errors. These errors are as follows: 

The correction for level error , designated by b (Art. 308), 
is ascertained from the readings of the striding-level. The 
value of a division of the level in seconds of time must have 
been previously ascertained by means of a level-trier. The 
effect of the level error is greatest at the zenith and diminishes 
to zero at the horizon. The correction in seconds of time is 
given (see formula (131)) by the following equation: 

bB — b cos (0 — S) sec <5 .( 1 62)' 

When the declination is north, it is to be regarded as 
having a plus sign for upper and a minus sign for lower 
culmination. When south it is negative. 

The correction for inequality of pivots can be made a part 
of the level correction. 


RED UCTION OF TIME OBSER VA 7'IONS. 


7 53 


the inequality of pivots; 

inclination of axis given by level for clamp west; 
inclination of axis given by level for clamp east; 
true inclination of axis for clamp west; 
true inclination of axis for clamp east;—then 

B' - B 

P — ~ ..(i^3) 

b — B -f -p for clamp west; 
b' — B' — p for clamp east. 

The correction for error of collimation , designated by c 
(Art. 308), is the departure of the mean of the threads from 
the optical axis of the telescope. For stars at upper culmi¬ 
nation with clamp west it is plus when the mean of the threads 
is east of the axis, and minus when it is west of it. For stars 
at lower culmination the reverse is the case. The value of c 
is one-half the difference between the clock error indicated by 
stars observed before and after reversal of the instrument, 
divided by the mean secant of the declinations of the stars. 
Th is is slightly complicated with the azimuth, although the 
effect of that is largely eliminated by the proper selection of 
stars. Consequently it is to be obtained by approximations, 
in conjunction with the azimuth errors. The correction to 
be applied to each star is, from formula (132), 

cC — c sec d,.(164) 

which is plus for a star at upper culmination, and minus for a 
star at lower culmination. It is least for equatorial stars and 
increases with the secant of the declination. 

The correction for deviation in azimuth , designated by a 
(Art. 308), represents the error in the setting of the instru¬ 
ment in the meridian. Its effect is zero at the zenith and in¬ 
increases towards the horizon. Since the instrument is liable 
to be disturbed during the operation of reversal, it is neces- 


Let p — 
B = 
B’ = 
b = 
b' =- 



754 


LONGITUDE . 


sary to determine the azimuth error separately, both before 
and after reversal. A comparison of the clock error, deter¬ 
mined from observations upon north and south stars, will fur¬ 
nish the data necessary for the determination of azimuth. 
?ractically, it is determined by elimination from equations 
involving the mean of all these stars observed in each of the 
two positions of the instrument, after correcting for level, and 
as it is slightly complicated with collimation it must be 
reached by two or more approximations. The errror is essen¬ 
tially positive when the telescope points east of south, and 
negative when west of south. The correction applicable to 
any star is expressed (see formula (130)) in the following 
equation : 

aA = a sin (0 — 6 ) sec < 5 . . (165) 

It must be understood that the declination when north is 
positive for upper and negative for lower culmination, and 
that with south declination it is negative. 

The right ascension of stars, as taken from the Star Cata¬ 
logue, must be corrected for diurnal aberration , which equals 
o s .02 I cos 0 sec 6 . This correction is positive for upper and 
negative for lower culmination. 

The foregoing corrections are summarized (see formula 
(135)) in the following equation: 

AT=a-(T 0 + aA + bB + cC). . . (166) 

A, B, C are constants, depending upon the latitude of the 
place of observation and the declination of the star. Tables 
for these quantities will be found in an appendix to Annual 
Report U. S. Coast and Geodetic Survey for 1880. (Extract 
reprinted herewith, Table LXX.) 

333 * Record of Time Observations. —On pages 755 to 
757 is an example of the form for record of observation and 
reduction of time observations, taken from a series made for 
the determination of position of Rapid, South Dakota. 


RECORD OF TIME 


OBSER VA TIONS. 


755 


1 r 




(A 

U ' 
> 

V 

^4-t 

O 

c 

cl 

V 

s 




















































































EXAMPLE OF RECORD OF TIME OBSERVATIONS— Continued. 


756 


LONGITUDE, 




div. 

- Q2C S - 

Mean of levels = —X .118 = — .027 = b. Inequality of pivots 











































































LONGITUDE—EXAMPLE OF REDUCTION 


757 



bD 

G 

O 

CO 

CO 



T 3 

c 

cfl 

rrj 

V) 

c 

s 

j3 

O 

u 

E 

o 

U 

v*-* 

C/3 

c 

_o 

rt 

cr 

Q-» 

bfi 

c 

a 

V- 

O 

£ 


<L> 

> 

ctf 

•G 

(U 

£ 


cfl 

a 

C/3 

u 


.c 

£ 

C/3 

a 

lx 

<u 

♦-* 

-G 

3 


N 

G 

bn 

c 


o 

c 

tuo 


a 

o 


bjc 

G 
—* 
U 
rt 

4 -* 

X) 

3 

C /3 


mo 

VO M M O 

^ O 0 o 


VO 

co 


+ 


CO 

vo 

m 

VO 

l ^ 


o o 


O o* 
O' lO 

O VO 
co co 


vO Ov 
N VO 

N 0 
1 + 
O O 


O N 

H C"> 
C". 

*» 

vo vo 

CO CO 


§ § 
N 4 


Cf 0 


vN vC n -*t- 

N C2 ^ ° 

" | T *2 0 

W 1 e. 


00 

CO 

o 


+ 

II 


CO LO . 

1 M CO ^ 

+ o 


«N^ 

+i 


II a 
o 

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+X^< 




+ 


o M IOVO ^ 
W M CO O 

II o ^ o 


S+++I 

“o II II II II 




5n 


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VO 

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+ 


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<LI <U 

ag- 

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>a 'O 
<< 


! ++l 

XXX 

oo O' 0 
goo 

l-H- 


moo 
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^ ^ ■n- 

Ct vo vo 
, • r- 

^ t^a 

rn^% 

"++£ 
« • 
C C> Ojs 

S ro co tj- 
E • o m 

• 3 ++S 

n ^ V) ^ 

i; ' • I 

•c H—h 

lit II 

^ ^ ^ 
O 00 00 ^ 

^ Cl N 

- • o 


3 <3 <\ 

CT* O 
*"« 

13 

E 

L. 

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£ 


vo 

0 


s 0 0 
o II II 

* Ov CO 

I 9 ^ 

|| vo 

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q.vg Ov 
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< I 4 - 


4 ?m n . 
o P 

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m | +oo« 

ro 1 1 ^ 0 


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kg " 
a£ 

<1 fc 


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w O' 

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» *; cs 

«■ “ ^ 

0 4- c/3 
1 c 

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m 

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+ 


2. 


0 0 0 


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vo vo 
00 co 

. * 

1 vo vo 
CO co 


w 13 I I 

JT^S 

• • ^ H H 

"+1+1 
! H 2 

£ § <5 

O 

VO • — 

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+ 


nl 

3 

cr 

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Cuo 

c 


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m 

* + 


Vj « 
00 
co 

CO 

+ 


o 

[Xh 





































































































758 


LONGITUDE , 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 

•(Extracted from Appendix 14, U. S. Coast and Geodetic Survey Report for 1880.) 

To find A enter left-hand column with the zenith distance; its intersection with declina¬ 
tion column given azimuth factor. 

To find B enter right-hand column with the zenith distance; its intersection with declina* 
tion column gives level factor. 

C is given on last line of each section of the table. 

Azimuth factor A = sin £ sec 5 . Star’s declination ± 5 . Inclination factor B = cos £ sec 8. 



o° 

IO° 

15 ° 

20° 

22° 

24 0 

26° 

28° 

3 °° 

32 0 

34 ° 

36° 

<r 

1° 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

I .02 

.02 

.02 

.02 

.02 

89° 

2 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

88 

3 

•05 

•05 

•05 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

87 

4 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

86 

5 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

. 10 

. IO 

. 10 

.IO 

.11 

85 

6 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

• T 3 

•13 

84 

7 

.12 

.12 

•13 

•13 

•13 

•13 

.14 

.14 

.14 

.14 

•15 

• 15 

83 

8 

.14 

.14 

.14 

• 15 

•15 

•15 

.16 

.16 

.16 

.16 

•17 

•17 

82 

q 

.l6 

1 . l6 

.16 

•17 

•17 

•17 

•17 

.18 

.18 

.18 

.19 

.19 

81 

10 

.17 

j .18 

.18 

.19 

.19 

.19 

.19 

.20 

.20 

1 

.21 

.21 

.21 

80 

11 

.19 

.19 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

•23 

•23 

.24 

79 

12 

.21 

.21 

.22 

.22 

.22 

•23 

.23 

.24 

.24 

.25 

.25 

.26 

78 

13 

.22 

•23 

•23 

•24 

.24 

•25 

•25 

.26 

.26 

.27 

.27 

.28 

77 

M 

.24 

•25 

•25 

.26 

.26 

.27 

•27 

.27 

.28 

.29 

.29 

•30 

76 

15 

.26 

.26 

•27 

.28 

.28 

.28 

.29 

.29 

•30 

• 3 i 

• 3 i 

•32 

75 

16 

.28 

.28 

.29 

.29 

•30 

•30 

• 3 i 

•31 

•32 

•33 

• 33 

•34 

74 

17 

.29 

•30 

•30 

• 3 i 

•31 

•32 

•33 

•33 

•34 

•34 

•35 

•36 

73 

18 

.31 

•31 

•32 

•33 

•33 

•33 

•34 

•35 

• 36 

•36 

• 37 

.38 

72 

19 

•33 

•33 

•34 

•35 

•35 

•36 

•36 

•37 

.38 

.38 

•39 

.40 

7i 

20 

•34 

•35 

•35 

.36 

•37 

•37 

.38 

•39 

.40 

.40 

.41 

.42 

70 

21 

•36 

•36 

•37 

.38 

•39 

•39 

.40 

.41 

.41 

.42 

•43 

•44 

69 

22 

•37 

.38 

•39 

.40 

.40 

.41 

.42 

.42 

•43 

•44 

•45 

.46 

68 

23 

•39 

.40 

.41 

.42 

.42 

•43 

• 44 

•44 

•45 

.46 

■47 

.4S 

67 

24 

.41 

.41 

.42 

•43 

•44 

•45 

•45 

.46 

•47 

.48 

•49 

• 50 

66 

25 

.42 

•43 

•44 

•45 

.46 

.46 

•47 

.48 

•49 

• 50 

• 5 i 

•52 

65 

26 

• 44 

•45 

•45 

•47 

•47 

.48 

•49 

•50 

• 5 i 

• 52 

•53 

• 54 

64 

27 

•45 

.46 

•47 

.48 

•49 

•50 

• 5 i 

• 51 

•52 

•54 

•55 

.56 

63 

28 

•47 

.48 

•49 

• 50 

• 5 i 

• 5 i 

• 52 

•53 

•54 

•55 

•57 

• 58 

62 

29 

.48 

•49 

• 50 

• 52 

•52 

•53 

• 54 

•55 

•56 

•57 

•58 

.60 

61 

30 

•50 

• 5 i 

• 52 

•53 

•54 

• 55 

• 56 

•57 

• 58 

•59 

.60 

.62 

60 

3 i 

.52 

• 52 

•53 

•55 

.56 

•56 

• 57 

•58 

•59 

.61 

.62 

.64 

59 

32 

•53 

• 54 

• 55 

• 56 

•57 

.58 

•59 

.60 

.61 

•63 

.64 

• 65 

58 

33 

•54 

• 55 

•56 

• 58 

•59 

.60 

.61 

.62 

•63 

.64 

.66 

.67 

57 

34 

.56 

•57 

•58 

•59 

.60 

.61 

.62 

•63 

• 65 

.66 

.67 

.69 

56 

35 

•57 

• 58 

•59 

.61 

.62 

•63 

.64 

.65 

.66 

.68 

.69 

• 71 

55 

36 

•59 

.60 

.61 

•63 

.63 

.64 

•65 

.67 

.68 

.69 

• 7 i 

•73 

54 

37 

.60 

.61 

.62 

.64 

• 65 

.65 

.67 

.68 

.70 

•7i 

•73 

•74 

53 

38 

.62 

•63 

.64 

.66 

.66 

.67 

.69 

.70 

• 7 i 

•73 

•74 

.76 

52 

39 

•63 

.64 

•65 

•67 

.68 

.69 

.70 

• 7 i 

•73 

•74 

.76 

•78 

5i 

40 

.64 

.65 

.66 

.68 

.69 

.70 

•72 

•73 

•74 

.76 

•77 

•79 

50 

4 i 

.66 

.67 

.68 

.70 

•71 

•72 

•73 

•74 

.76 

•77 

•79 

.81 

49 

42 

.67 

.68 

.69 

• 7 i 

•72 

•73 

•74 

.76 

•77 

•79 

.81 

•83 

48 

43 

.68 

.69 

• 7 i 

•73 

•74 

• 75 

.76 

•77 

•79 

.80 

.82 

.84 

47 

44 

.69 

• 7 i 

•72 

•74 

• 75 

.76 

•77 

•79 

.80 

.82 

.84 

.86 

46 

45 

•7i 

.72 

•73 

• 75 

.76 

•77 

•79 

.80 

.82 

.83 

•85 

•87 

45 


























































A, B, C STAB FACTORS , 


759 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


£ 

o° 

io° 

15 0 

20 ° 

22 ° 

24 0 

26° 

28° 

3 o° 

32 0 

34 ° 

3 6° 1 

£ 

46° 

.72 

•73 

•74 

•77 

.78 

•79 

.80 

. 82 

.83 

•85 

•87 

.89 

44 ° 

47 

•73 

• 74 

• 76 

•78 

•79 

.80 

.81 

• 83 

. 84 

.86 

.88 

.90 

43 

48 

•74 

.76 

• 77 

•79 

. 80 

.81 

.83 

. 84 

.86 

.88 

.90 

.92 

42 

49 

•75 

•77 

•78 

.80 

.81 

•83 

.84 

.86 

.87 

.89 

.91 

•93 

4 i 

50 

•77 

.73 

•79 

.82 

.83 

.84 

.85 

.87 

.89 

. 90 

.92 

•95 

40 

5 i 

1 

•73 

•79 

. 80 

.83 

.84 

•85 

•87 

.88 

.90 

.92 

•94 

.96 

39 

52 

•79 

.80 

.82 

. 84 

.85 

.86 

.88 

.89 

.91 

•93 

•95 

•97 

38 

53 

.80 

.81 

•83 

.85 

.86 

.87 

.89 

.91 

.92 

•94 

.96 

•99 

37 

54 

.81 

.82 

.84 

.86 

.87 

.89 

.90 

.92 

•93 

•95 

•98I 

I .OO: 

36 

55 

.82 

•83 

.85 

•87 

.88 

. 90 

.91 

•93 

•95 

•97 

•99 

1.01 

35 

56 

•33 

. 84 

.86 

.88 

. 89 

.91 

.92 

•94 

.96 

.98 

1.00 

1.02 

34 

57 

.84 

.85 

.87 

. 89 

.90 

.92 

•93 

•95 

•97 

•99 

1.01 

1.04 

33 

53 

•85 

.86 

.88 

.90 

.91 

•93 

• 94 

.96 

.98 

1.00 

1.02 

1.05 

32 

59 

.86 

•87 

. 89 

.91 

.92 

•94 

•95 

•97 

•99 

1.01 

1.03 

1.06 

3 i 

60 

• 87 

.88 

.90 

.92 

•93 

•95 

.96 

.98 

1.00 

1.02 

1.04 

O 

M 

30 

61 

• 87 

.89 

.91 

•93 

• 94 

.96 

•97 

•99 

1.01 

1.03 

1.05 

I.OS 

29 

62 

.88 

.90 

.91 

•94 

•95 

•97 

.98 

1.00 

1.02 

1.04 

1.06 

I .09 

28 

63 

.89 

.91 

.92 

•95 

.96 

.98 

•99 

1.01 

1.03 

1.05 

1.07 

I . IO 

27 

64 

.90 

.91 

•93 

.96 

•97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

I . II 

26 

65 

.91 

.92 

•94 

.96 

.98 

•99 

1 

1.01 

1.03 

1.05 

1.07 

1.09 

I . 12 

25 

66 

.91 

•93 

•95 

•97 

•99 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

I-I 3 

24 

67 

.92 

•94 

•95 

.98 

•99 

1.01 

1.02 

1.04 

1.06 

1.09 

1.11 

1.14 

23 

68 

•93 

• 94 

.96 

•99 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.12 

115 

22 

69 

•93 

•95 

•97 

•99 

1.01 

1.02 

1.04 

1.06 

1 .c8 

1.10 

i-i 3 

1.15 

21 

70 

•94 

•95 

•97 

1.00 

1.01 

1.03 

1.05 

1.06 

1.09 

1.11 

1 .13 

1.16 

20 

7 i 

•95 

.96 

.98 

1.01 

1.02 

1 

I .04 

1.05 

1.07 

1.09 

1.12 

1.14 

i-i 7 

19 

72 

•95 

•97 

.98 

1.01 

1.03 

1.04 

1.06 

1.08 

1.10 

1.12 

1 .15 

l -'l 

18 

73 

.96 

•97 

•99 

1.02 

1.03 

1.05 

1.06 

1.08 

1.10 

I-I 3 

1 .15 

1.18 

17 

74 

.96 

.98 

1.00 

1.02 

r.04 

1.05 

1.07 

1.09 

1.11 

I -13 

1.16 

1.19 

16 

75 

•97 

.98 

1.00 

1.03 

1.04 

1.06 

I 

1.08 

1.09 

1.12 

1.14 

1.16 

1.19 

15 

76 

•97 

.•99 

1.00 

1.03 

1.05 

1 

1.06 

1.08 

1.10 

1.12 

1.14 

1.17 

1.20 

14 

77 

•97 

•99 

1.01 

1.04 

1.05 

1.07 

1.08 

1.10 

1 • I 3 

1 .15 

1.17 

1.20 

13 

73 

.98 

•99 

1.01 

1.04 

1.05 

1.07 

1.09 

1.11 

I-I 3 

115 

1.18 

1.21 

12 

79 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.09 

1.11 

1 .13 

1.16 

1.18 

1.21 

11 

80 

.98 

1.00 

1.02 

1.05 

1.06 1.08 

1 

1.10 

I. 12 

1.14 

1.16 

1.19 

1.22 

10 

81 

•99 

1.00 

r.02 

1.05 

1.07 1.08 

1.10 

1.12 

1.14 

1.17 

1.19 

1.22 

9 

82 

•99 

1.01 

1.03 

1-05 

1.07 1.08 

1.10 

1.12 

1.14 

1.17 

1.19 

1.22 

8 

83 

• 99 

1.01 

1.03 

1.06 

1.07 1.09 

1.10 

1.12 

115 

1.17 

1.20 

1.23 

7 

84 

•99 

1.01 

1.03 

1 1.06 

1.07 1.09 

1.11 

I-I3 

1 .15 

1.17 

1.20 

1.23 

6 

85 

1.00 

1.01 

1.03 

1.06 

1.0} 

1.09 

1.11 

!..3 

1 • 15 

1.17 

1.20 

1.23 

5 

86 

1.00 

1.01 

1.03 

1.06 

1.08 i.og 

I 1.11 

; 1.13 

1 .15 

1.18 

1.20 

1.23 

4 

87 

1.00 

1.01 

1.03 

1.06 

1.08 i.og 

1.11 

1.13 

I-I 5 

1. iS 

1.20 

1.23 

3 

88 

1.00 

1.01 

1-03 

1.06 

1.08 i.og 

1.11 

1 113 

1 • 1 5 

1.18 

1.20 

1.23 

2 

89 

1.00 

1.02 

1.04 

1.06 

1.0S i.og 

1.11 

i-i 3 

i-i 5 

1.18 

1.21 

1.24 

1 

90 

1.00 

1.02 

1.04 

J 1.06 

1 

1.08 i.og 


1 13 

1 .15 

| 1. iS 

1.21 

1.24 

0 













































































y 6 o 


LONGITUDE , 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


<; 

38° 

40° 

4 i° 

42 0 

43 ° 

44 ° 

45 ° 

46° 

47 ° 

4 8° 

49 ° 

50° 

i 

i° 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

•03 

.03 

•03 

•03 

89 ° 

2 

.04 

.05 

•05 

.05 

•05 

.05 

•05 

.05 

.05 

•05 

•05 

-05 

88 

3 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

87 

4 

.09 

.09 

.09 

.09 

. 10 

. IO 

. 10 

. 10 

. IO 

. 10 

. II 

. II 

£6 

5 

.11 

.11 

.11 

. 12 

. 12 

.12 

. 12 

•13 

•13 

•13 

•13 

•13 

85 

6 

•13 

.14 

.14 

.14 

.14 

•15 

•15 

•15 

•15 

. 16 

• l6 

. l6 

84 

7 

.15 

. 16 

. l6 

. 16 

•17 

•17 

•17 

. 18 

. l8 

. 18 

.19 

.19 

83 

8 

. l8 

. 18 

. l8 

.19 

.19 

.19 

.20 

. 20 

.20 

. 21 

.21 

.22 

82 

9 

.20 

.20 

.21 

.21 

.21 

. 22 

.22 

.22 

•23 

•23 

. 24 

. 24 

81 

10 

. 22 

.23 

•23 

•23 

.24 

•24 

•25 

•25 

•25 

.26 

.26 

•27 

80 

ii 

. 24 

.25 

•25 

.26 

'. 26 

.27 

•27 

.28 

.28 

.28 

.29 

•30 

79 

12 

. 26 

• 27 

• 27 

.28 

.28 

.29 

. 29 

•30 

•30 

• 3 i 

•32 

•32 

78 

13 

. 29 

. 29 

•30 

•30 

• 3 i 

•31 

•32 

•32 

•33 

•34 

•34 

•35 

77 

14 

•31 

•32 

•32 

•33 

•33 

•34 

• 34 

•35 

•35 

•36 

•37 

.38 

76 

15 

•33 

•34 

•34 

•35 

•35 

•36 

•37 

•37 

.38 

1 -39 

•39 

.40 

75 

16 

•35 

•36 

•37 

•37 

•38 

•38 

•39 

.40 

.40 

.41 

.42 

•43 

74 

17 

•37 

•33 

•39 

•39 

.40 

.41 

.41 

.42 

• 43 

•44 

•45 

•45 

73 

18 

•39 

.40 

.41 

.42 

.42 

•43 

• 44 

•44 

•45 

.46 

• 47 

.48 

72 

19 

.41 

.42 

•43 

• 44 

•45 

•45 

.46 

•47 

. 48 

• 49 

•50 

• 5 i 

7 i 

20 

•43 

•45 

•45 

.46 

• 47 

.48 

.48 

•49 

•50 

• 5 i 

• 52 

•53 

70 

21 

• 45 

• 47 

•47 

. 48 

•49 

• 50 

•51 

• 52 

• 52 

•54 

•55 

•56 

69 

22 

.48 

•49 

•50 

•50 

• 5 i 

• 52 

•53 

•54 

•55 

.56 

• 57 

•58 

68 

23 

•50 

• 5 i 

•52 

•53 

•53 

•54 

•55 

.56 

•57 

• 58 

.60 

.6l 

67 

24 

•52 

•53 

• 54 

•55 

.56 

• 57 

•58 

•59 

.60 

.61 

.62 

•63 

66 

25 

•54 

•55 

•56 

•57 

•58 

•59 

.60 

.61 

.62 

•63 

.64 

. 66 

65 

26 

.56 

• 57 

•53 

•59 

.60 

.61 

.62 

•63 

.64 

.65 

.67 

.68 

64 

27 

• 53 

•59 

.60 

* 

.61 

.62 

•63 

.64 

.65 

.67 

.68 

.69 

• 7 i 

63 

28 

.60 

.61 

.62 

•63 

. 64 

.65 

.66 

.68 

.69 

.70 

•72 

•73 

62 

29 

.61 

.63 

.64 

•65 

.66 

.67 

.69 

.70 

• 7 i 

.72 

•74 

•75 

61 

3 C 

•63 

•65 

.66 

.67 

.68 

.69 

• 7 i 

•72 

•73 

•75 

.76 

•78 

60 

3 i 

•65 

.67 

.68 

.69 

.70 

• 72 

•73 

•74 

•75 

•77 

1 78 

. 80 

59 

32 

.67 

.69 

.70 

•7i 

.72 

• 74 

•75 

.76 

•78 

•79 

.81 

.82 

58 

33 

.69 

•71 

•72 

•73 

•74 

.76 

•77 

•78 

. 80 

.81 

.83 

•85 

57 

34 

• 7 i 

•73 

•74 

• 75 

.76 

• 78 

•79 

. 80 

.82 

. 84 

•85 

.87 

56 

35 

•73 

75 

.76 

•77 

•78 

. 80 

.81 

.83 

.84 

.86 

.87 

.89 

55 

36 

•75 

•77 

•78 

•79 

. 80 

. 82 

.83 

•85 

.86 

.88 

.90 

.91 

54 

37 

.76 

•79 

.80 

.81 

.82 

.84 

.85 

•87 

.88 

. 90 

.92 

•94 

53 

3 S 

.78 

.80 

.82 

.83 

. 84 

.86 

•87 

.89 

. 90 

.92 

•94 

. 96 

52 

39 

. 80 

. 82 

.83 

• 85 

.86 

•87 

.89 

.91 

. 92 

•94 

.96 

.98 

5 i 

40 

.82 

.84 

.85 

.86 

.88 

.89 

.91 

•93 

•94 

.96 

.98 

1.00 

50 

4 i 

•83 

.86 

.87 

.88 

.90 

.91 

•93 

•94 

.96 

.98 

1.00 

1.02 

49 

42 

.85 

• 87 

.89 

.90 

.91 

•93 

•95 

.96 

.9S 

1.00 

1.02 

1.04 

48 

43 

.86 

.89 

.90 

.92 

•93 

•95 

. 96 

.98 

1.00 

1.02 

1.04 

1.06 

47 

44 

.89 

.90 

.92 

•93 

•95 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

46 

45 

.90 

.92 

•94 

•95 

•97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

45 

J 









































































A, B, C STAR FACTORS. 


761 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


i 

38° 

40° 

4i° 

42 ° 

43 ° 

44 ° 

45 ° 

46 ° 

47 ° 

48 ° 

49 ° 

50° 

£ 

46° 

• 9 i 

• 94 

•95 

•97 

.98 

1.00 

1.02 

1,04 

1.05 

1.07 

I.IO 

1.12 

44 ° 

47 

•93 

•95 

•97 

.98 

I.OO 

1.02 

1.03 

105 

I 07 

1.09 

I.II 

1.14 

43 

48 

• 94 | 

•97 

.98 

1.00 

1.02 

I.03 

1.05 

1.07 

I.09 

1.11 

I -13 

1.16 

42 

49 

.96 

•99 

1.00 

1.02 

I.O3 

1.05 

1.07 

1.09 

1.11 

1 .13 

I -15 

1.17 

4 i 

50 

•97 

1.00 

1.01 

1.03 

1.05 

1.06 

I.08 

1.10 

1.12 

1.14 

1.17 

1.19 

40 

5 i 

•99 

1.01 

1.03 

1-05 

1.06 

1.08 

I.IO 

1.12 

1.14 

1.16 

1.18 

1.21 

39 

52 

1.00 

1.03 

1.04 

1.06 

I.OS 

I. IO 

I. II 

113 

I -15 

1.18 

1.20 

1.23 

38 

53 

1.01 

1.04 

1.06 

1.07 

I.09 

I. II 

i-i 3 

I-I 5 

i.iy 

1.19 

1.22 

1.24 

37 

54 

1.03 

1.06 

1.07 

1.09 

I. II 

1.12 

1.14 

1.16 

1.19 

1.21 

I.23 

1.26 

36 

55 

1.04 

1 07 

1.08 

1.10 

1.12 

1.14 

1.16 

1.18 

1.20 

1.22 

1-25 

1.27 

35 

50 

1.05 

1.08 

I.IO 

1.12 

II 3 

I -15 

1.17 

1.19 

1.22 

1.24 

1.26 

1.29 

34 

57 

1.06 

1.09 

1.11 

1.13 

I -15 

1.17 

1.19 

1.21 

1.23 

1.25 

1.28 

1. 3 i 

33 

53 

1.08 

1.11 

1.12 

1.14 

1.16 

1.18 

1.20 

1.22 

1.24 

1.27 

I.29 

1.32 

32 

59 

1.09 

1.12 

1.14 

1.15 

I-I 7 

1.19 

1.21 

1.23 

1.26 

1.28 

i- 3 i 

i -33 

3 i 

60 

I. IO 

i-i 3 

i-i 5 

1.17 

1.18 

1.20 

1.22 

1.25 

1.27 

1.29 

1.32 

i -35 

30 

61 

I. II 

1.14 

1.16 

1.18 

1.20 

1.22 

1.24 

1 26 

1.28 

i- 3 i 

i -33 

1.36 

29 

62 

1.12 

1 .15 

i-i 7 

1.19 

1.21 

1.23 

1.25 

1.27 

1.29 

1.32 

1-35 

i -37 

28 

63 

1.13 

1.16 

1.18 

1.20 

1.22 

I.24 

1.26 

1.28 

i- 3 i 

1-33 

1.36 

1-39 

27 

64 

1.14 

1.17 

1.19 

1 21 

1.23 

1.25 

1.27 

1.29 

1.32 

i -34 

i -33 

1.40 

26 

•65 

1 • 1 5 

1.18 

1.20 

1.22 

1.24 

1.26 

1.28 

1.30 

i -33 

1-35 

1.38 

I.41 

25 

66 

1.16 

1.19 

1.21 

1.23 

1.25 

1.27 

1.29 

1.32 

i -34 

i *37 

i -39 

1.42 

24 

67 

1.17 

1.20 

1.22 

1.24 

1.26 

1.28 

1.30 

i -33 

1-35 

1.38 

1.40 

i -43 

23 

68 

1.18 

1.21 

1.23 

1-25 

1.27 

1.29 

131 

1-33 

1.36 

i -39 

1.41 

1.44 

22 

69 

1.18 

1.22 

1.24 

1.26 

r.28 

1.30 

1.32 

1-34 

i -37 

1.40 

1.42 

1.45 

21 

70 

1.19 

1.23 

1.25 

1.26 

1.28 

I. 3 I 

i -33 

1-35 

1.38 

1.40 

1-43 

1.46 

20 

71 

1.20 

1.23 

r. 25 

1.27 

1.29 

I-31 

1-34 

1.36 

i -39 

1.41 

1.44 

1.47 

* 19 * 

72 

1.21 

1.24 

1.26 

1.28 

1.30 

1.32 

1-34 

1-37 

i -39 

1.42 

1-45 

1.48 

18 

73 

I. 21 

1.25 

1.27 

1.29 

1.31 

i -33 

1-35 

1.38 

1.40 

1-43 

1.46 

1.49 

17 

74 

1.22 

1-25 

1.27 

1.29 

I - 3 I 

i -34 

1.36 

1.38 

1.41 

1.44 

1.46 

1.49 

16 

75 

1.23 

1.26 

1.28 

1.30 

1.32 

1-34 

i -37 

i -39 

1.42 

1.44 

1.47 

1.50 

15 

7 6 

1.23 

1.27 

1.29 

1.31 

1-33 

i -35 

i -37 

1.40 

1.42 

1-45 

1.48 

1.5i 

14 

77 

1.24 

1.27 

1.29 

I- 3 1 

1-33 

i -35 

1.38 

1.40 

i -43 

1.46 

1.48 

1-52 

13 

73 

1.24 

1.28 

1.30 

1.32 

i -34 

1.36 

1.38 

1.41 

1.43 

1.46 

1.49 

1.52 

12 

79 

1.25 

1.28 

1.30 

1.32 

1-34 

1.36 

i -39 

1.41 

1.44 

1.47 

1.50 

1-53 

11 

80 

1.25 

1.29 1.30 

i -33 

i -35 

i -37 

1-39 

1.42 

1.44 

1.47 

1.50 

1-53 

10 

81 

1.25 

1.29 

i *3 c 

i -33 

1-35 

i -37 

1.40 

1.42 

i -45 

1.48 

i-5i 

i -54 

9 

82 

1.26 

1.29 1 .31 

i -33 

i -35 

1.38 

1.40 

i -43 

1-45 

1.48 

1.5i 

i -54 

8 

83 

1.26 

I.30 

1-32 

1-34 

1.36 

1.38 

1.40 

i -43 

1.46 

1.48 

1. 5 i 

1 54 

7 

84 

1.26 

I.30 

1.32 

1-34 

i. 3 6 

1.38 

1.41 

i -43 

1.46 

1.49 

1.52 

1-55 

6 

35 

1.26 

I.30 

1.32 

1-34 

1.36 

| 1-38 

1.41 

i -43 

1.46 

1.49 

1.52 

1-55 

5 

86 

1.27 

I.30 

1.32 

1-34 

1.36 

i -39 

1.41 

1.44 

1.46 

1.49 

1.52 

1-55 

4 

87 

1.27 

1.30 

1.32 

1-34 

1-37 

i -39 

1.41 

1.44 

1.46 

1.49 

1.52 

1 55 

3 

88 

1.27 

1.30 

1.32 

1-34 

1-37 

I i -39 

1.41 

1.44 

1.46 

1.49 

1.52 

1 • 5 5 

2 

89 

1.27 

1.31 

1.32 

T *35 

i -37 

i -39 

1.41 

1.44 

1.47 

1.49 

1.52 

1.56 

I 

A 

90 

1.27 

i- 3 i 

1.32 

I 1 - 35 

1-37 

i -39 

1.41 

1.44 

1.47 

1.49 

1.52 

1 5 o 

U 

_ 






































































762 


LONGITUDE. 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


i 


52 0 

53 ° 

54 ° 

55 ° 

56° 

57 ° 

58° 

59 ° 

6 o° 

6 oi ° 

6 i° 

£ 

1° 

.03 

.03 

•03 

.03 

•03 

.03 

•03 

•03 

•03 

•03 

.04 

.04 

89 ° 

2 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.07 

.07 

.07 

.07 

.07 

SS 

3 

.08 

.08 

.09 

.09 

.09 

.08 

.IO 

. IO 

.10 

. IO 

•II 

.11 

87 

4 

.11 

.11 

.12 

.12 

.12 

.12 

•13 

• 1 3 

•14 

.14 

.14 

.14 

86 

5 

.14 

.14 

.14 

•15 

•15 

.16 

. l6 

. 16 

•17 

•17 

.18 

.18 

85 

6 

•17 

• 17 

•17 

.18 

.18 

.19 

.19 

.20 

.20 

.21 

.21 

.22 

84 

7 

.19 

.20 

.20 

.21 

.21 

.22 

.22 

•23 

.24 

.24 

•25 

•25 

83 

8 

.22 

•23 

•23 

•24 

.24 

.25 

.26 

.26 

•27 

.23 

.28 

.29 

82 

9 

•25 

• 25 

.26 

.26 

•27 

.28 

. 29 

•29 

•30 

•31 

•32 

•32 

Sr 

10 

.28 

,28 

•29 

•30 

• 3 ° 

•31 

•32 

•33 

•34 

•35 

•35 

•36 

80 

11 

•30 

•31 

•32 

•32 

•33 

• 34 

•35 

.36 

•37 

•38 

•39 

•39 

79 

12 

•33 

•34 

•35 

•35 

.36 

•37 

.38 

•39 

.40 

• 42 

.42 

•43 

78 

13 

•36 

•36 

•37 

.38 

•39 

.40 

.41 

.42 

•44 

•45 

.46 

.46 

77 

14 

•38 

•39 

.40 

.41 

.42 

.43 

• 44 

.46 

•47 

.48 

49 

•50 

76 

15 

.41 

.42 

•43 

• 44 

• 45 

.46 

.43 

•49 

•50 

•52 

•53 

■53 

75 

16 

• 44 

• 45 

.46 

•47 

.48 

•49 

• 51 

•52 

• 54 

• 55 

•56 

•57 

74 

17 

.46 

• 47 

•49 

• 50 

• 51 

.52 

•54 

•55 

•57 

.58 

•59 

.60 

73 

18 

•49 

• 50 

• 5 i 

•53 

•54 

.55 

•57 

•58 

.60 

.62 

•63 

.64 

72 

19 

•52 

•53 

•54 

• 55 

• 57 

.58 

.60 

.61 

•63 

.65 

.66 

.67 

7 i 

20 

•54 

• 56 

•57 

• 58 

.60 

.61 

•63 

.64 

.66 

.63 

.69 

.70 

70 

21 

•57 

• 58 

•59 

.61 

.62 

.64 

.66 

.68 

.70 

•72 

•73 

• 74 

69 

22 

.60 

.61 

.62 

.64 

• 65 

.67 

.69 

•71 

•73 

•75 

.76 

•77 

68 

23 

.62 

•63 

.65 

.66 

.68 

.70 

• 72 

•74 

.76 

•78 

•79 

.81 

67 

24 

•65 

.66 

.68 

.69 

• 71 

•73 

• 75 

•77 

•79 

.81 

•83 

.84 

66 

25 

.67 

.69 

.70 

• 72 

•74 

.76 

•78 

.80 

.82 

.85 

.86 

•87 

65 

26 

• 70 

• 7 i 

•73 

• 75 

.76 

.78 

.80 

•83 

.85 

.88 

.S9 

.90 

64 

27 

•72 

•74 

•75 

•77 

•79 

.81 

.83 

.86 

.88 

.91 

.92 

•94 

63 

28 

•75 

.76 

•78 

.80 

.82 

. 84 

.86 

.89 

.91 

•94 

•95 

•97 

62 

29 

•77 

•79 

.81 

.82 

.84 

• 87 

.89 

.91 

•94 

•97 

.98 

1.00 

61 

30 

•79 

.81 

•83 

•85 

•87 

.89 

.92 

•94 

•97 

1.00 

I.OT 

1.03 

60 

3 i 

.82 

.84 

.86 

.88 

.90 

.92 

•95 

•97 

1.00 

1.03 

1.05 

1.06 

59 

32 

.84 

.86 

.88 

.90 

.92 

•95 

•97 

1.00 

1.03 

1.06 

1.08 

1.09 

58 

33 

.87 

.88 

.91 

•93 

•95 

•97 

1.00 

1.03 

1.06 

1.09 

I. II 

1.12 

57 

34 

.89 

.91 

•93 

•95 

•97 

1.00 

1.03 

1.05 

1.09 

1.12 

I. 14 

1 .15 

56 

35 

.91 

•93 

•95 

.98 

1.00 

1.03 

1.05 

1.08 

1.11 

i -15 

I.l6 

1.18 

55 

36 

9-3 

•95 

.98 

1 00 

1.03 

1.05 

1.08 

1.11 

1.14 

1.18 

1.19 

1.21 

54 

37 

.96 

.98 

1.00 

1.02 

1.05 

1.08 

I. IO 

1.14 

T * 17 

1.20 

1.22 

1.24 

53 

38 

.98 

1.00 

1.02 

1.05 

1.07 

1.10 

i-i 3 

1.16 

1.20 

1.23 

1.25 

1.27 

52 

39 

1.00 

1.02 

1.05 

1.07 

1.10 

1.12 

1 .15 

1.19 

1.22 

1.26 

1.28 

1.30 

5 i 

40 

1.02 

1.04 

1.07 

1.09 

1.12 

1.15 

1.18 

1.21 

1-25 

1.29 

I - 3 I 

i -33 

50 

41 

1.04 

1.07 

1.09 

1.12 

1.14 

1.17 

1.20 

1.24 

1.27 

i- 3 i 

i -33 

i -35 

49 

42 

1.06 

1.09 

1.11 

1.14 

1.17 

1.20 

1.23 

1.26 

I.30 

i -34 

1.36 

1.38 

48 

43 

1.08 

1.11 

1 .13 

1.16 

1.19 

1.22 

1.25 

1.29 

1.32 

1.36 

i -39 

1.41 

47 

44 

1.10 

i-i 3 

1 .15 

1.18 

I.21 

1.24 

1.28 

i- 3 i 

T «35 

1-39 

1.-41 

1 43 

46 

45 

1.12 

1 .15 

117 

1.20 

1.23 

1.26 

1.30 

T *33 

i -37 

1.41 

1.44 

1.46 

45 















































































A, B, C STAR FACTORS. 


7 6 3 


* 


Table LXX. 


FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 



5 i° 

5 *° 

53 ° 

54 ° 

55 ° 

56° 

57 ° 

58° 

59 ° 

6o° 

604 0 

6i° 

£ 

46° 

1.14 

1.17 

1.19 

1.22 

1.25 

1.29 

1.32 

1.361 

I.40 

I.44 

1.46 

I.48 

44 " 

47 

1.16 

1.19 

I.21 

1.24 

1.27 

i- 3 i 

i -34 

1-38 

I.42 

I.46 

1.49 

1 .51 

43 

48 

1.18 

1.21 

1.23 

1.26 

I.30 

i -33 

1-36 

1.40 

I.44 

I.48 

1.50 

1-53 

42 

49 

1.20 

1-23 

1.25 

1.28 

1.32 

1-35 

i -39 

1.42 

1.47 

I -51 

i -53 

1.56 

4 i 

50 

1.22 

I.24 

1.27 

! 

1.30 

1-34 

i -37 

1 41 

1.44 

I.49 

i -53 

1.56 

1.58 

40 

5 1 

1.23 

1.26 

1.29 

I 32 

i -35 

i -39 

1-43 

1 47 

1.51 

1-55 

1.58 

1.60 

39 

52 

1.25 

1.28 

1.31 

1-34 

i -37 

1.41 

i -45 

1.49 

i -53 

t .58 

1.60 

1.63 

38 

53 

1.27 

1.30 

1 • 33 1 

1.36 

1-39 

i -43 

i -47 

i- 5 i 

1-55 

1.60 

1.62 

1.65 

37 

54 

I.29 

i- 3 1 

i -34 

1.38 

1.41 

i -45 

1.49 

1-53 

i -57 

1.62 

1.64! 

1.671 

36 

55 

1.30 

1-33 

*- 3 & 

1 39 

1-43 

1.46 

1.50 

i -55 

i -59 

1.641 

1 . 66 j 

1.69 

35 

56 

1.32 

i -35 

I. 38 1 

1.41 

J .45 

1.48 

*•52 

1.56 

1.61 

1.66 

1.68 

1.71 

34 

57 

i -33 

1.36 

i -39 

1-43 

1.46 

1.50 

1-54 

1.58 

1.63 

1.68 

1.70 

i -73 

33 

58 

i -35 

1.38 

1.41 

1-44 

1 4S 

1.52 

1.56 

1.60 

1-651 

1.70 

1.72 

1 • 7 5 j 

32 

59 

i- 3 6 

i -39 

1.42 

1.46 

1.49 

i -53 

1-57 

1.62 

1.66 

1. 7 i 

1.74 

1.77 

3 i 

60 

1.38 

1.41 

1.44 

1 47 

.. 5 i 

1-55 

i -59 

1.63 

1.68 

i -73 

1.76 

1.79 

30 

61 

1-39 

1.42 

I 45 

1.49 

1.53 

1.56 

1.61 

1.65 

1.70 

T -75 

1.78 

1.So 

29 

62 

1.40 

143 

1 1-47 

1.50 

1-54 

1.58 

1.62 

1.67 

1 71 

I 1-77 

| i -79 

1.82 

28 

63 

1.42 

1-45 

1.49 

1.52 

i -55 

1-59 

1.64 

1.68 

i -73 

1.78 

1.8. 

1.84 

27 

64 

i -43 

1.46 

1-49 

1-53 

1 57 

1.61 

1 65 

1.70 

1-75 

1.80 

1.83 

1.85 

26 

65 

1.4a 

1.47 

1. 5 i 

i -54 

1.58 

1.62 

1.66 

1.71 

1.76 

1.81 

, 1.84 

1.87 

25 

66 

1-45 

1.48 

1.52 

1 55 

1-59 

1.63 

1.68 

1.72 

1-77 

1.83 

1.85 

1.88 

24 

67 

1.46 

1.50 

i -53 

1-57 

1.60 

1.65 

1.69 

1-74 

i -79 

1.84 

1.87 

1.90 

23 

68 

1.47 

1. 5 i 

154 

1.58 

1.62 

1 1 66 

1.70 

i -75 

1.80 

1.85 

1.88 

1.91 

22 

69 

1.48 

1 52 

i -55 

1-59 

1.63 

1.67 

1.71 

1.76 

1.81 

1.87 

1.90 

i -93 

21 

70 

1 49 

i -53 

1.56 

1.60 

1.64 

1.68 

1.73I 1.77 

1.82 

1.88 

1.91 

1.94 

20 

7 1 

1.50 

i -54 

i -57 

1.61 

1.65 

1.69 

1 74 

1.78 

1.84 

1.89 

1.92 

1.05 

19 

72 

1. 5 i 

1-54 

1.58 

1.62 

1.66 

1.70 

1-75 

1.80 

1.85 

1.90 

i -93 

1.96 

18 

73 

1-52 

1-55 

i -59 

1.63 

1.67 

1 71 

1.76 

1.80 

!.86 

1.91 

1.94 

1.97 

17 

74 

i -53 

1.56 

1.60 

1-63 

1.68 

1.72 

1.76 

' 1.81 

1.87 

1.92 

i .95 

1.98 

16 

75 

1-53 

i -57 

1.60 

1 1.64 

1.68 

1.73 

1.77 

1.82 

1.88 

j 

1-93 

1.96 

1.99 

15 

76 

i -54 

1.58 

1.61 

: I.6«? 

1.69 

i -73 

1.78 

1.83 

' 1.88 

1.94 

1.97 

2.00 

14 

77 

1-55 

1.58 

1.62 

1.66 

1.70 

1 74 

1 79 

1.84 

1.89 

i -95 

1.9S 

2.01 

13 

78 

i -55 

i -59 

1.62 

1.66 

1.70 

i -75 

1.80 

1.85 

1 90 

1.96 

1.99 

2.02 

12 

79 

1.56 

i -59 

1.63 

1.67 

1.71 

1.76 

1.80 

1.85 

1.91 

1.96 

1.99 

2.02 

11 

80 

1.56 

1.60 

1.64 

1.67 

1.72 

1.76 

1.81 

1.86 

1.91 

1.97 

2.00 

2.03 

10 

81 

1-57 

1.60 

1.64 

1.68 

1.72 

1.77 

1.81 

1.86 

1.92 

1.98 

2.01 

2.04 

9 

82 

1-57 

1.61 

1.64 

1.68 

i -73 

1-77 

1.82 

r.87 

1.92 

1.98 

2.01 

2.04 

8 

83 

1.58 

1.61 

1.65 

1.69 

i -73 

1-77 

1.82 

1.87 

1-93 

1.99 

2.02 

2.05 

7 

84 

1.58 

1.62 

1.65 

1.69 

i -73 

1.78 

1.83 

1.88 

1-93 

1.99 

2.02 

2.05 

6 

85 

1.58 

1.62 

1-65 

1.69 

1.74 

1.78 

1.83 

1 1.88 

i -93 

1.99 

2.02 

2.05 

5 

86 

1.59 

1.62 

1.66 

i. 70 

1.74 

1.78 

1.83 

1.88 

1.94 

2.00 

2.03 

2.06 

1 4 

87 

1.^9 

1.62 

1.66 

1.70 

1.74 

1 79 

1.83 

1 1.8S 

1.94 

2.00 

2.03 

2.06 

3 

88 

1-59 

1.62 

1.66 

1.70 

i -74 

1.79 

1.83 

1.89 

1.94 

2.00 

2.03 

2.06 

2 

89 

1-59 

1.62 

1.66 

1.70 

1.74 

i -79 

1.84 

! 1.89 

1.94 

2.00 

2.03 

2.06 

I 

90 

..59 

1.62 

1.66 

1 

| 1.70 

1.74 

i -79 

1.84 1.89 

1 

1 i -94 

2.00 

2.03 

2.06 

°J 























































































76 4 


LONGITUDE. 



Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


i 

61 h° 

62° 

62*° 

63° 

631° 

6 4 ° 

64 ° 

65° 

Io 

b 32 

66° 

66 h° 

6 7 ° 

£ 

V 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

1 eg- 

2 

.07 

•07 

.08 

.08 

.08 

.08 

.08 

.08 

• OS 

.09 

.09 

.og 

88 

3 

.n 

.11 

.11 

.12 

.12 

.12 

.12 

.12 

•13 

•13 

•13 

•13 

87 

4 

• I5 

•15 

• 15 

• 15 

. 16 

. 16 

. 16 

•17 

•17 

• I7 

.18 

.18 

86 

5 

.18 

.19 

.19 

.19 

.20 

| 

.20 

. 20 

.21 

.21 

.21 

.22 

.22 

85 

6 

.22 

.22 

•23 

•23 

•23 

•24 

.24 

•25 

•25 

.26 

.26 

•27 

84 

7 

.26 

.26 

.26 

•27 

•27 

.28 

.28 

.29 

.29 

■30 

•31 

• 3 i 

s 3 

8 

• 29 

•30 

•30 

•31 

• 3 i 

•32 

.32 

•33 

•34 

•34 

•35 

.36 

82 

9 

•33 

•33 

•34 

.35 

•35 

•36 

.36 

•37 

.38 

•39 

•39 

.40 

81 

10 

•36 

•37 

•38 

•38 

•39 

.40 

.40 

.41 

.42 

•43 

•43 

•44 

80 

ii 

.40 

.41 

.41 

.42 

•43 

•44 

• 44 

•45 

.46 

•47 

.48 

•49 

79 

12 

■44 

•44 

•45 

.46 

•47 

•47 

.48 

•49 

•50 

• 5 i 

•52 

•53 

78 

13 

•47 

.48 

•49 

•50 

• 50 

• 5 i 

•52 

•53 

•54 

•55 

.56 

• 58 

77 

14 

• 5 i 

•52 

•52 

•53 

•54 

•55 

• 56 

•57 

• 58 

•59 

.61 

.62 

76 

15 

•54 

•55 

•56 

•57 

• 58 

• 59 

.60 

.61 

.62 

'.64 

.65 

.66 

75 

16 

•58 

•59 

.60 

.61 

.62 

•63 

.64 

.65 

.66 

.68 

.69 

• 71 

74 

17 

.61 

.62 

•63 

.64 

.66 

.67 

.68 

.69 

.70 

.72 

•73 

• 75 

73 

18 

• 65 

.66 

.67 

.68 

.69 

.70 

• 72 

• 73 

•74 

.76 

• 77 

•79 

72 

19 

.68 

.69 

.70 

•72 

•73 

•74 

.76 

•77 

•78 

.80 

.82 

.83 

7 i 

20 

•72 

•73 

•74 

•75 

• 77 

•79 

•79 

.81 

•83 

. 84 

.86 

.88 

70 

21 

• 75 

.76 

• 78 

•79 

.80 

.82 

•83 

•85 

.86 

.88 

.90 

.02 

69 

22 

•78 

.80 

.81 

.82 

.84 

• 85 

.87 

.89 

.90 

.92 

•94 

.96 

68 

23 

.82 

•83 

.85 

.86 

.88 

.89 

.91 

.92 

•94 

.96 

.98 

I.GO 

67 

24 

• 85 

•87 

.88 

.90 

.91 

•93 

•94 

.96 

.98 

1.00 

1.02 

I.04 

66 

25 

.89 

.90 

.92 

•93 

•95 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

I.OS 

65 

26 

.92 

•93 

•95 

•97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

I.IO 

1.12 

64 

27 

•95 

•97 

.98 

1.00 

1.02 

1.04 

1.05 

1.07 

1.09 

1.12 

1.14 

1.16 

63 

28 

.98 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

1 .13 

1 .15 

1.18 

1.20 

62 

29 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

Ti 3 

i-i 5 

i-i 7 

1.19 

1.22 

I.24 

61 

30 

1.05 

1.07 

1.08 

1.10 

1.12 

1.14 

1.16 

1.18 

1.21 

1.23 

1-25 

1.28 

60 

3 i 

1.08 

1.10 

1.11 

i-i 3 

I-I 5 

1 .17 

1.20 

1.22 

1.24 

1.27 

1.29 

1.32 

59 

32 

1.11 

1.13 

1 .15 

1.17 

1.19 

1.21 

1.23 

1.25 

1.28 

1.30 

1.33 

1.36 

58 

33 

1.14 

1.16 

1.18 

1.20 

1.22 

1.24 

1.26 

1.29 

I. 3 I 

i -34 

i -37 

1.39 

57 

34 

1 • 1 7 

i* 19 

1.21 

1.23 

1.25 

1.27 

1.30 

1.32 

i -35 

1-37 

1.40 

1-43 

56 

1 35 

1.20 

1.22 

1.24 

1.26 

1.29 

i- 3 i 

i -33 

1.36 

1.38 

1.41 

1.44 

I.47 

55 

36 

1.23 

1.25 

1.27 

1.30 

1.32 

1-34 

i -37 

i -39 

1.42 

i -45 

4*47 

I -51 

54 

37 

1.26 

1.28 

1.30 

i -33 

i -35 

i -37 

1.40 

1.42 

i -45 

1.48 

1. 5 i 

1-54 

53 

33 

1.29 

1. 3 1 

1-33 

1.36 

1.38 

1.40 

i -43 

1.46 

1.48 

1. 5 i 

r -54 

1.58 

52 

39 

1.32 

i -34 

1 36 

i -39 

1.41 

1-43 

1.46 

1.49 

1.52 

i -55 

1.58 

1.61 

5 i 

40 

i -35 

1-37 

i -39 

1.42 

1.44 

1.47 

1.49 

1.52 

i -55 

1.58 

1.61 

1.65 

50 

4 i 

i -37 

1.40 

1.42 

1-45 

1.47 

1.50 

i -53 

i -55 

1.58 

1.61 

1.64 

1.68 

49 

42 

1.40 

1.42 

1-45 

1.47 

1.50 

i -53 

i -55 

1.58 

1.61 

1.64 

1.68 

1. 7 i 

48 

43 

1-43 

1 - 45 , 

1.48 

1.50 

i -53 

1.56 

1.58 

1.61 

1.64 

1.68 

1.71 

1-75 

47 

44 

1.46 

1.48 

1.50 

i -53 

1 - 56 , 

1.58 

1.61 

1 64 

1.67 

i- 7 i 

1.74 

1.78 

46 

45 

1.48 

1. 5 i 

i -53 

1.56 

1.58 

1.61 

1.64 

1.67 

1.70 

1.74 

i -77 

1.81 

45 


















































































A, B, C STAB FACTORS, 


765 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


£ 

6i*° 

62° 

62i° 

63° 

63P 

6 4 ° 

64 5 ° j 

65° 

65 h ° 

66° 

66^» 

67° 

i 

46° 

i • 5 1 

i -53 

1.56 

1.58 

1.61 

1.64 

1.67 

1.70 

1.74 

1 • 77 

1.80 

1.84 

44 ° 

47 

i -53 

1.55 

1.58 

1.61 

1.64 

1.67 

1.70 

i -73 

1.76 

1.80 

1.83 

1.87 

43 

48 

i -55 

1-58 

1.60 

1.63 

1.66 

1.69 

1.72 

i -75 

1.79 

1.82 

1.86 

1.90 

42 

49 

1. 58 

1.61 

1.63 

1.66 

1.69 

1.72 

i -75 

1.79 

1.82 

i. 86 

1.89 

1-93 

4 i 

50 

1.60 

1.63 

i. 66 

1.69 

1 • 7 2 

1-75 

1.78 

1.81 

1.85 

1.88 

1.92 

1.96 

40 

5 i 

1.63 

1.66 

1.68 

1 • 7 i 

1.74 

1.77 

1.80 

1.84 

1.87 

1.91 

1 95 

1.99 

39 

52 

1.65 

1.68 

1.71 

1.74 

1.77 

1.80 

1.83 

1.86 

1.90 

1.94 

1.98 

2.02 

38 

53 

1.67 

1.70 

i -73 

1.76 

1.79 

1.82 

1.85 

1.89 

i -93 

1.96 

2.00 

2.04 

37 

54 

1.69 

1.72 

1 -75 

1.78 

1.81 

1.85 

1.88 

1.91 

i -95 

1.99 

2.03 

2.07 

36 

55 

1.72 

1.74 

1.77 

1.80 

1.84 

1.87 

1.90 

1.94 

1.98 

2.01 

2.05 

2.10 

35 

56 

1.74 

r.77 

1.80 

1.83 

1.86 

1.89 

i -93 

1.96 

2.00 

2.04 

2.08 

2.12 

34 

57 

1.76 

1.79 

1.82 

1.85 

1.88 

1.91 

i -95 

1.98 

2.02 

2.06 

2.10 

2.15 

33 

58 

1.78 

1.81 

1.84 

1.87 

1.90 

i -93 

i -97 

2.01 

2.05 

2.08 

2.13 

2.17 

32 

59 

1.80 

1.83 

1.86 

1.89 

1.92 

1-95 

i -99 

2.03 

2.07 

2.11 

2.15 

2.19 

3 i 

60 

1.81 

1.84 

1.88 

1.91 

1.94 

1.97 

2.01 

2.05 

2.091 

2.13 

2.17 

2.22 

30 

61 

1.83 

1.86 

1.89 

i -93 

1.96 

2.00 

2.03 

2.07 

2. iii 

2.15 

2.19 

2.24 

29 

62 

1.85 

r. 88 

1.91 

1.94 

1.98 

2.01 

2.05 

2.09 

2.13 

2.17 

2.21 

2.26 

28 

63 

1.87 

1.90 

i -93 

1.96 

2.00 

2.03 

2.07 

2.11 

2.15 

2.19 

2.23 

2.28 

27 

64 

1.88 

1.91 

i -95 

1.98 

2.02 

2.05 

2.09 

2.13 

2.17 

2.21 

2.25 

2.30 

26 

65 

1.90 

i -93 

1.96 

2.00 

2.03 

2.07 

2.11 

2.14 

2.19 

2.23 

2.27 

2.32 

25 

66 

1.91 

1.95 

1.98 

2.01 

2.05 

2.08 

2.12 

2.16 

2.20 

2.25 

2.29 

2-34 

24 

67 

1.93 

1.96 

1.99 

2.03 

2.06 

2.10 

2.14 

2.18 

2.22 

2.26 

2.31 

2.36 

23 

68 

1.94 

1.97 

2.01 

2.04 

2.08 

2.11 

2.15 

2.19 

2.24 

2.28 

2.32 

2-37 

22 

69 

1.96 

1.99 

2.02 

2.06 

2.09 

2.13 

2.17 

2.21 

2.25 

2.30 

2.34 

2-39 

21 

70 

1.97 

2.00 

2.03 

2.07 

2.11 

2.14 

2.18 

2.22 

2.27 

2.31 

2.36 

2.40 

20 

7 i 

1.98 

2.01 

2.05 

2.08 

2.12 

2.16 

2.20 

2.24 

2.28 

2.32 

2-37 

2.42 

19 

72 

1.99 

2.03 

2.06 

2.09 

2.13 

2.17 

2.21 

2.25 

2.29 

2-34 

2.38 

2-43 

18 

73 

2.00 

2.04 

2.07 

2.11 

2.14 

2.18 

2.22 

2.26 

2.31 

2-35 

2.40 

2-45 

17 

74 

2.01 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.32 

2.36 

2.41 

2.46 

16 

75 

2.02 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.29 

2-33 

2.37 

2.42 

2.47 

15 

76 

2.03 

2.07 

2.10 

2.14 

2.17 

2.21 

2.25 

2.30 

2-34 

2-39 

2-43 

2.48 

14 

77 

2.04 

2.07 

2.11 

2.15 

2.18 

2.22 

2.26 

2.31 

2-35 

2.40 

2.44 

2.49 

13 

78 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.31 

2.36 

2.40 

2-45 

2.50 

12 

79 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2-37 

2.41 

2.46 

2.51 

11 

80 

2.06 

2.10 

2.13 

2.17 

2.21 

2.25 

2.29 

2.33 

2.38 

2.42 

2.47 

2.52 

10 

8, 

2.07 

2.10 

2.14 

2.18 

2.21 

2.25 

2.29 

2-34 

2.38 

2.43 

2.48 

2.53 

9 

82 

2.08 

2.11 

2.15 

2.18 

2.22 

2.26 

2.30 

2-34 

2-39 

2-43 

2.48 

2-53 

8 

83 

2.08 

2.12 

2.15 

2.19 

2.22 

2.26 

2.31 

2-35 

2-39 

2.44 

2.49 

2-54 

7 

84 

2.08 

2.12 

2.15 

12.19 

2.23 

2.27 

2.31 

2-35 

2.40 

2-45 

2.49 

2-55 

6 

85 

2.09 

2.12 

2.16 

2.19 

2.23 

2.27 

2.31 

2.36 

2.40 

2-45 

2.50 

2-55 

5 

86 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

2.41 

2.45 

2.50 

2-55 

4 

87 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

2.41 

2.46 

2.50 

2.56 

3 

88 

i 2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

2.41 

2.46 

2.51 

2.56 

2 

89 

2.10 

2.13 

2.17 

2.20 

2.24 

2.28 

2.32 

2-37 

2.41 

2.46 

2.51 

2.56 

1 

90 

2.10 

2.13 

2.17 

2.20 

2.24 

2.28 

1 

2.32 

2.37 

1 

2.41 

2.46 

2.51 

2.56 

0 



























































































766 


LONGITUDE. 


Table LXX. 


FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


£ 

1 67* 0 

68 ° 

682° 

69° 

69^ 

70° 

O 

O 

702 0 

0 

OW 

O 

71 ° 

7 * 1 ° 

7 i 2 ° 

£ 

1° 

.05 

•05 

.05 

•05 

•05 05 

•05 

•05 

• 05 

•05 

•05 

•05 

89° 

2 

.09 

.09 

. 10 

. 10 

I . IO 

j . IO 

. IO 

. IO 

. 11 

. II 

. II 

. II 

88 

3 

.14 

.14 

.14 

•15 

•15 

I - x 5 

•15 

. 16 

. 16 

. l6 

. l6 

. 16 

87 

4 

.18 

.19 

.19 

. 20 

. 20 

. 20 

. 21 

. 21 

.21 

. 21 

. 22 

. 22 

86 

5 

•23 

•23 

•24 

.24 

•25 

.25 

. 26 

. 26 

. 26 

.27 

•27 

• 27 

85 

6 

.27 

.28 

. 28 

. 29 

•30 

.31 

•31 

• 3 i 

•32 

•32 

•33 

•33 

84 

7 

•32 

•33 

•33 

•34 

•35 

•36 

.36 

•37 

•37 

•37 

.38 

•38 

83 

8 

•36 

•37 

•38 

•39 

.40 

.41 

.41 

.42 

. 42 

•43 

•43 

•44 

82 

9 

.41 

.42 

•43 

•44 

•45 

.46 

.46 

•47 

•47 

.48 

•49 

•49 

81 

10 

•45 

.46 

•47 

•49 

•50 

• 5 i 

•51 

•52 

•53 

•53 

•54 

•55 

80 

ii 

•50 

• 5 i 

• 52 

•53 

•54 

•56 

•56 

•57 

•58 

•59 

•59 

. 60 

79 

12 

•54 

•56 

•57 

.58 

•59 

.61 

.62 

. 62 

•63 

. 64 

•65 

.66 

78 

13 

•59 

.60 

.61 

.63 

.64 

.66 

.67 

.67 

.68 

. 69 

.70 

•71 

77 

14 

•63 

•65 

.66 

.68 

. 69 

• 7 i 

•72 

• 72 

•73 

•74 

•75 

.76 

76 

IS 

.68 

.69 

•71 

.72 

•74 

.76 

•77 

•78 

.78 

: 79 

. 80 

.81 

75 

16 

•72 

•74 

•75 

•77 

•79 

.81 

.82 

.83 

. 84 

•85 

.86 

.87 

74 

17 

.76 

• 78 

.80 

.81 

•83 

•85 

.86 

.88 

.89 

.90 

.91 

.92 

73 

18 

.81 

.83 

. 84 

.86 

.88 

.90 

.91 

•93 

•94 

•95 

.90 

•97 

72 ! 

19 

.85 

.87 

.89 

.91 

•93 

•95 

.96 

.98 

•99 

1.00 

1.01 

1.03 

7 i ! 

20 

.89 

.91 

•93 

•95 

.98 

1.00 

1.01 

1.02 

1.04 

1.05 

1.06 

1.08 

70 

21 

•94 

.96 

.98 

1.00 

1.02 

1.05 

1.06 

1.07 

1.09 

I. TO 

1.11 

I-I 3 

69 

22 

.98 

1.00 

1.02 

1.05 

1.07 

1.09 

1.11 

I. T 2 

1.14 

I I 5 

1.17 

1.18 

68 

23 

1.02 

1.04 

1.07 

1.09 

1.12 

1.14 

1.16 

I.I7 

1.19 

1.20 

1.21 

1.23 

67 

2 4 

1.06 

1.09 

1.11 

1.14 

1.16 

1.19 

1.20 

I . 22 

1.23 

1.25 

1.27 

1.28 

66 

25 

1.10 

113 

115 

1. iS 

1.21 

1.24 

1-25 

1.27 

1.28 

I.30 

1 • 3 1 

i -33 

65 

26 

i -15 

i-i 7 

I . 2 C 

1.22 

1.25 

1.28 

1.30 

1 • 3 1 

i -33 

i -35 

1.36 

1.38 

64 

27 

1.19 

1.21 

I . 24 

1.27 

1.30 

i -33 

i -34 

1.36 

1.38 

i -39 

1.41 

i -43 

63 

28 

1.23 

1.25 

1.28 

1 • 3 1 

i -34 

i -37 

i -39 

1.41 

1.42 

1.44 

1.46 

1.48 

62 

29 

1.27 

1.29 

I.32 

i -35 

1 38 

1.42 

i -43 

i -45 

1.47 

1.49 

1. 5 i 

1-53 

61 i 

30 

1 • 3 1 

i -33 

I.36 

i -39 

1-43 

1.46 

1.48 

1.50 

1.52 

i -54 

1.56 

1.58 

60 

3 i 

i -35 

1.38 

I .40 

1.44 

1.47 

i- 5 i 

1.52 

i -54 

1.56 

1.58 

1.60 

1.62 

59 

32 

i -39 

1.42 

I.45 

1.48 

i- 5 i 

i -55 

i -57 

i -59 

1.61 

1.63 

1.65 

1.67 

58 

33 

1.42 

i -45 

I.49 

1.52 

i -55 

i -59 

1.61 

1.63 

1.65 

1.67 

1.69 

1.72 

57 

34 

1.46 

1.49 

i -53 

1.56 

1.60 

1.63 

1.65 

1.68 

1.70 

1.72 

1.74 

1.76 

56 

35 

1.50 

i -53 

1-56 

1.60 

1.64 

1.68 

1.70 

1.72 

1.74 

1.76 

1.78 

1.81 

55 

36 

1-54 

i -57 

1.60 

1.64 

1.68 

1.72 

1.74 

1.76 

1.78 

1.80 

1.83 

1.85 

54 

37 

1-57 

1.6t 

1.64 

1.68 

1.72 

1.76 

1 • 78 

1.80 

1.83 

1.85 

1.87 

1.90 

53 

38 

1.61 

1.64 

1.68 

1.72 

1.76 

1.80 

1.82 

1.84 

1.87 

1.89 

1.91 

1.94 

52 

39 

1 -651 

1.68 

1.72 

i -75 

1.80 

1.84 

1.86 

1.88 

1.91 

1-93 

1.96 

1.98 

5 i 

40 

1.68 

1.72 

i -75 

1.79 

1.84 

1.88 

1.90 

i -93 

1-95 

1.97 

2.00 

2.03 

50 

4 i 

1.71 

i -75 

1.79 

1.83 

1.87 

1.92 

1.94 

1.96 

1.99 

2.01 

2.04 

2.07 

49 

42 

1-75 

1.79 

1.83 

1.87 

1.91 

1.96 

1.98 

2.00 

2.03 

2.05 

2.08 

2.11 

48 

43 

1.78 

1.82 

1.86 

1.90 

i -95 

1.99 

2.02 

2.04 

2.07 

2.09 

2.12 

2.15 

47 

44 

1. S2 

1.85 

1.90 

1.94 

1.98 

2.03 

2.06 

2.08 

2.11 

2.13 

2.16 

2.19 

46 

45 

1.85 

1.89 

i -93 

1.97 

2.02 

2.07 

2.09 

2.12 

2.14 

2.17 

2.20 

2.23 

45 















































































A, B, C STAR FACTORS. 


76 7 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


i 

6 1 \° 

68° 

68j° 

6g° 

69* 0 

70 ° 

704 0 

7 °i° 

O 

nyfi 

0 

71 0 

- T lo 

7 U 

712° 

< 

46° 

1.88 

I.92 

I.96 

2.01 

2.05 

2.10 

2.13 

2.15 

2.18 

2.21 

2.24 

2.27 

44 ° 

47 

1.91 

i -95 

2.00 

2.O4 

2.09 

2.14 

2.16 

2. 19 

2.22 

2.25 

2.27 

2.30 

43 

48 

1.94 

1.98 

2.02 

2.07 

2.12 

2.17 

2.19 

2.22 

2.25 

2.28 

2.31 

2 34 

42 

49 

1.97 

2.01 

2.06 

2.11 

2.16 

2.21 

2.23 

2.26 

2.29 

2.32 

2.351 

2.38 

j l j 

50 

2.00 

2.04 

2.O9 

2.14 

2.19 

2.24 

2.27 

2.29 

2.32 

2.35 

2.38 

2.41 

40 

5 i 

2.03 

2.07 

2. 12 

2.17 

2.22 

2.27 

2.30 

2-33 

2.36 

2-39 

2.42 

2-45 

39 

52 

2.06 

2.10 

2.15 

2.20 

2.25 

2.3O 

2-33 

2.36 

239 

2.42 

2-45 

2.48 

38 

53 

2.09 

2.13 

2. l8 

2.23 

2.28 

2-33 

2.36 

2-39 

2.42 

2-45 

2.48 

2.52 

37 

54 

2.11 

2.16 

2.21 

2.20 

2.31 

2-37 

2-39 

2.42 

2.45 

2.48 

2 5 2 

2-55 

36 

55 

2.14 

2.19 

2.23 

2.29 

2.34 

2.4O 

2.42 

2.45 

2.48 

2.52 

2.55 

2 58 

35 

56 

2.17 

2.21 

2.26 

2.3I 

2.37 

2.42 

2-45 

2.48 

2.51 

2-55 

2.5S 

2.61 

34 

57 

2.19 

2.24 

2 29 

2-34 

2-39 

2.45 

2.48 

2.51 

2.54 

2.58 

2.61 

2.64 

33 

58 

2.22 

2.26 

2.32 

2 37 

2.42 

2.48 

2.51 

2-54 

2.57 

2.61 

2.64 

2.67 

32 

59 

2.24 

2.29 

2-34 

2-39 

2-45 

2.51 

2-54 

2-57 

2.60 

2.63 

2.67 

2.70 

3 i 

60 

2.26 

2.3I 

2.36 

2.42 

2.47 

2-53 

2.56 

2.59 

2.63 

2.66 

2.69 

2-73 

30 

61 

2.29 

2 33 

2-39 

2.44 

2.50 

2.56 

2-59 

2.62 

2.65 

2.69 

2.72 

2.76 

29 

62 

2.31 

2.36 

2.41 

2.46 

2.52 

2.58 

2.61 

2.O4! 

2.68 

2.71 

2.75 

2.78 

28 

^3 

2-33 

2.38 

2-43 

2.49 

2-54 

2.60 

2.64 

2.67 

2.70 

2.74 

2.77 

2.S11 

27 

64 

2-35 

2.40 

2 45 

2.51 

2 57 

2.63 

2.66 

2.69 

2.73 

2.76 

2.So 

2.83 

26 

65 

2-37 

2.42 

2.47 

2-53 

2-59 

2.65 

2.6S 

2.71 

2.75 

2.78 

2.82 

2.86 

25 

66 

2-39 

2.44 

2.49 

2-55 

2.61 

2.67 

2.70 

2.74 

2.77 

2.81 

2.84 

2.88 

24 

67 

2.41 

2.46 

2.51 

2-57 

2.63 

2.69 

2.72 

2.76 

2.79 

2.83 

2 86 

2.90 

23 

68 

2.42 

2-47 

2-53 

2-59 

2.65 

2.7I 

2.74 

2.78 

2.81 

2.85 

2.88 

2.92 

22 

69 

2-44 

2 49 

2.55 

2.61 

2.67 

2-75 

2.76 

2.SO 

2.83 

2 87 

2.90 

2.94 

21 

70 

2.46 

2.51 

2.56 

2.62 

2.68 

2-75 

2.78 

2.8l 

2.85 

2.89 

2.92 

2.96 

20 

7 i 

2.47 

2.52 

2.58 

2.64 

2.70 

2.77 

2.80 

2.83 

2.87 

2.90 

2.94 

2.98 

19 

72 

2.49 

2-54 

2-59 

2.65 

2.72 

2 78 

2.81 

2.85 

2.88 

2.92 

2.96 

3.00 

18 

73 

2.50 

2-55 

2.61 

2.67 

2-73 

2 .SO 

2.83 

2.86 

2.90 

2.94 

2 97 

3.01 

17 

74 

2.51 

2-57 

2.62 

2.68 

2.74 

2.8l 

2.84 

2.88 

2.92 

2-95 

2.99 

3-03 

16 

75 

2.52 

2.58 

2.64 

2.70 

2.76 

2.82 

2.86 

2.89 

2.93 

2.97 

3.00 

3-°4 

15 

76 

2.54 

2-59 

2.65 

2.71 

2.77 

2.84 

2.87 

2.91 

2.95 

2.99 

3.02 

3.06 

14 

77 

2-55 

2.60 

2.66 

2.72 

2.78 

2.85 

2.88 

2.92 

2.95 

2.99 

3-03 

3-07 

13 

78 

2.56 

2.61 

2.67 

2-73 

2.79 

2.86 

2.89 

2-93 

2.97 

3.00 

3-04 

3.08 

12 

79 

2-57 

2.62 

2.68 

2.74 

2.80 

2.87 

2.91 

2.94 

2.98 

3 02 

3-05 

3.09 

11 

80 

2-57 

2.63 

2.69 

2-75 

2.81 

2.88 

2.91 

2-95 

2.99 

3.02 

3.06 

3.10 

10 

81 

2.58 

2.64 

2.69 

2.76 

2.82 

2.89 

2.92 

2.96 

3.00 

3-03 

3-07 

3 11 

9 

82 

2-59 

2.64 2.70 

2.76 

2.83 

2.90 

2-93 

2.97 

3.00 

3-04 

3.08 

3*12 

8 

83 

2-59 

2.65 

2.71 

2.77 

2.83 

2.90 

2.94 

2.97 

3.01 

3-05 

3-09 

3 -i 3 

7 

84 

2.60 

2 66 

2.71 

2.78 

2.84 

2.91 

2.94 

2.98 

3.02 

3.06 

3.00 

3-13 

6 

85 

2.60 

2.66 

2.72 

2.78 

2.84 

' 2.91 

2-95 

2.98 

3.02 

3.06 

3.10 

3 .i 4 

5 

86 

2.61 

2 66 

2.72 

2.78 

2.85 

2.92 

2.95 

2.99 

3-03 

3.06 

3.10 

3-14 

4 

87 

2 61 

2.67 

2 72 

2.79 

2.85 

2.92 

2.95 

2.99 

303 

3-07 

3 - 11 

3-15 

3 

88 

2.61 

2.67 

2-73 

2.79 

2.S5 

2.92 

2.96 

2.99 

3-03 

3-07 

3 - 11 

3 15 

2 

89 

2.61 

2.67! 2.73 

2.79 

2.86 

2.92 

2.96 

3.00 

3-03 

3-07 

3 - 11 

3-15 

1 

90 

2.61 

2.67 

2-73 

2.79 

2.86 

2.92 

2.96 

3.00 

3-03 

3-07 

3 - 11 

3-15 

0 






















































































768 


LONGITUDE . 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 



0 

C 3 hj< 

72 0 

724° 

72 h ° 

7 2f° 

73 ° 

734 ° 

0 

i-*XN 

CT ) 

_ l 

73 l° 

74 ° 

0 

^|C 4 

| 

< 

1° 

•05 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

89 ° 

2 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.12 

.12 

•13 

•13 

88 

3 

•17 

• 17 

• 17 

•17 

.18 

.18 

• l8 

. iS 

.19 

.19 

.19 

87 

4 

.22 

•23 

.23 

•23 

•23 

.24 

.24 

•24 

•25 

•25 

.26 

86 

5 

.28 

.28 

.29 

.29 

.29 

•30 

•30 

•31 

•31 

•32 

•32 

85 

6 

•33 

•34 

• 34 

•35 

•35 

•36 

•36 

•37 

•37 

•38 

•39 

84 

7 

•39 

•39 

.40 

.41 

.41 

.42 

.42 

•43 

•44 

•44 

•45 

83 

8 

•44 

• 45 

.46 

.46 

•47 

.48 

.48 

•49 

• 50 

.50 

.51 

82 

9 

• 50 

*51 

• 5 i 

• 52 

•53 

•53 

•54 

•55 

.56 

•57 

• 58 

81 

10 

•55 

.56 

•57 

.58 

•59 

.60 

.60 

.61 

.62 

•63 

.64 

80 

11 

.61 

.62 

.63 

.63 

.64 

• 65 

.66 

.67 

.68 

.69 

• 70 

79 

12 

.66 

.67 

.68 

.69 

.70 

• 7 i 

• 72 

•73 

•74 

•75 

• 77 

78 

13 

•72 

•73 

•74 

• 75 

• 76 

• 77 

• 78 

•79 

.80 

.82 

.83 

77 

14 

• 77 

• 78 

•79 

.80 

.82 

.83 

.84 

•85 

•87 

.88 

.89 

76 

15 

•83 

.84 

•85 

.86 

.87 

.89 

.90 

• 9 i 

•93 

•94 

•95 

75 

16 

.88 

.89 

.91 

.92 

•93 

•94 

.96 

•97 

•99 

1.00 

1.02 

74 

17 

•03 

•95 

.96 

•97 

•99 

1.00 

r.ci 

1.03 

1.05 

1.06 

1.08 

73 

18 

•99 

1.00 

1.01 

1.03 

1.04 

1.06 

1.07 

1.09 

1.10 

1.12 

1.14 

72 

19 

1.04 

1.05 

1.07 

1.08 

1.10 

1.11 

i-i 3 

i-i 5 

1.16 

1.18 

1.20 

7 i 

20 

1.09 

1.11 

1.12 

1.14 

1 .15 

1.17 

1.19 

1.20 

1.22 

1.24 

1.26 

70 

21 

1.14 

1.16 

1.17 

1.19 

1.21 

1.22 

1.24 

1.26 

1.28 

1.30 

1.32 

69 

22 

1.20 

1.21 

1.23 

1.25 

1.26 

1.28 

1.30 

1.32 

1-34 

1.36 

1.38 

68 

23 

1-25 

1.26 

1.20 

1.30 

1.32 

i -34 

1 36 

1.38 

1.40 

1.42 

1.44 

67 

24 

1.30 

1.32 

J -33 

i -35 

1-37 

1-39 

1.41 

1-43 

1-45 

1.48 

1.50 

66 

25 

i -35 

1-37 

i -39 

1.41 

1.42 

i -45 

1-47 

1.49 

i- 5 i 

1-53 

1.56 

65 

26 

1.40 

1.42 

1.44 

1.46 

1.48 

1-50 

1.52 

i -54 

1-57 

i -59 

1.61 

64 

27 

1-45 

1.47 

1.49 

I - 5 I 

i -53 

1 .£5 

1.58 

1.60 

1.62 

1.65 

1.67 

63 

28 

1.50 

1.52 

1-54 

1.56 

1.58 

1.60 

1.63 

1.65 

1 68 

1.70 

i -73 

62 

29 

i -55 

1-57 

1-59 

1.61 

1.63 

1.66 

1.68 

1. 7 i 

1.78 

1.76 

1.79 

61 

30 

1.60 

1.62 

1.64 

1.66 

1.69 

1.71 

i -73 

1.76 

1.79 

1.81 

1.84 

60 

3 i 

1.64 

1.67 

1.69 

i- 7 i 

1 74 

1.76 

1.79 

1.81 

1.84 

1.87 

1.90 

59 

32 

1.69 

i- 7 i 

1.74 

1.76 

1.79 

1.81 

1.84 

1.87 

j .89 

1.92 

i -95 

58 

33 

1.74 

1.76 

1.79 

1.81 

1.84 

1.86 

1.89 

1.92 

i -95 

1.98 

2.01 

57 

34 

1.79 

I.8l 

1.83 

1.86 

1.89 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

56 

35 

1.83 

1.86 

1.8S 

1.91 

1-93 

1.96 

1.99 

2.02 

2.05 

2.08 

2.11 

55 

36 

1.88 

1.90 

i -93 

i -95 

1.98 

2.01 

2.04 

2.07 

2.10 

2.13 

2.16 

54 

37 

1.92 

1-95 

1.97 

2.00 

2.03 

2.06 

2.09 

2.12 

2.15 

2.18 

2.22 

53 

38 

1.97 

1.99 

2.02 

2.05 

2.08 

2.11 

2.14 

2.17 

2.20 

2.23 

2.27 

52 

39 

2.01 

2.04 

2.06 

2.09 

2.12 

2.15 

2.18 

2.22 

2.25 

2.2S 

2.32 

51 

40 

2.05 

2.08 

2.11 

2.14 

2.17 

2.20 

2.23 

2.26 

2.30 

2-33 

2-37 

50 

41 

2.09 

2.12 

2.15 

2.18 

2.21 

2.24 

2.28 

2.31 

2.34 

2.38 

2.42 

49 

42 

2.14 

2.16 

2.19 

2.22 

2.26 

2.29 

2.32 

2.36 

2-39 

2-43 

2.46 

48 

43 

2.18 

2.21 

2.24 

2.27 

2.30 

2 33 

2-37 

2.40 

2-44 

2.47 

2 - 5 i 

47 

44 

2.22 

2.25 

2.28 

2.31 

2 - 34 , 

2.38 

2.41 

2-45 

2.48 

2.52 

2.56 

46 

45 

2.26 

2.29 

2.32 

2-35 

2.38 

2.42 

2-45 

2-49 

2-53 

2.56 

2.60 

45 


























































A, S, C STAR FACTORS. 


769 


Table LXX. 


FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


r— ■ -- 

7’!° 

72 0 

72I 0 

7 2^° 

7 2 4° 

73° 

734° 

73i° 

73 3° 

74° 

74l° 

i 

46° 

2.30 

2-33 

2.36 

2-39 

2.42 

2.46 

2.49 

2-53 

2.57 

2.61 

2.65 

44 ° 

47 

2-33 

2.37 

2.40 

2-43 

2.47 

2.50 

2-54 

2-57 I 

2.61 

2.65 

2.69 

43 

48 

2.37 

2.40 

2-44 

2.47 

2.51 

2-54 

2.58 

2.62 

2.66 

2.70 

2.74 

42 

49 

2.41 

2-44 

2,48 

2.51 

2-55 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

4 i 

50 

2-45 

2.48 

2.51 

2-55 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

2.82 

40 

5 i 

2.48 

2.51 

2-55 

2.58 

2.62 

2.66 

2.70 

2-74 

2.78 

2.82 

2.86 

39 

52 

2.52 

2-55 

2.58 

2 62 

2.66 

2.69 

2-73 

2.77 

2.82 

2.86 

2.90 

38 

53 

2.55 

2.58 

2.62 

2.66 

2.69 

2-73 

2.77 

2.81 

2.85 

2.90 

2.94 

37 

54 

2.58 

2.62 

2.65 

2.69 

2-73 

2-77 

2.81 

2.85 

2.89 

2-94 

2.98 

3 ^ 

55 

2.62 

2.65 

2.69 

2.72 

2.76 

2.80 

2.84 

2.88 

2-93 

2.97 

3.02 

35 

56 

2.65 

2.68 

2.72 

2.76 

2.80 

2.84 

2.88 

2 92 

2.96 

3.01 

3-05 

3 4 

57 

2.68 

2.71 

2-75 

2-79 

2.83 

2.87 

2.91 

2-95 

3.00 

3-04 

3-09 

33 

58 

2.71 

2-74 

2.78 

2.82 

2.86 

2.90 

2.94 

2.99 

3-03 

3.08 

3.12 

32 

59 

2.74 

2.77 

2.81 

2.85 

2.89 

2-93 

2.97 

3.02 

3.06 

3 - 11 

3.16 

3 i 

60 

2.76 

2.80 

2 84 

2.88 

2.92 

2.96 

3.01 

3-05 

3-09 

3-14 

3 -i 9 

30 

61 

2.79 

2.83 

2.87 

2.91 

2-95 

2-99 

3-04 

3.08 

3-13 

3-17 

3.22 

29 

62 

2.82 

2.86 

2.90 

2.94 

2.98 

3.02 

3.06 

3 - 11 

3.16 

3.20 

3-25 

2S 

63 

2.84 

2.88 

2.92 

2.96 

3.00 

3-05 

3.09 

3 - f 4 

3.18 

323 

3.28 

27 

64 

2.87 

2.91 

2-95 

2.99 

3 03 

3-07 

3.12 

3.16 

3 21 

3.26 

3 - 3 i 

26 

65 

2.89 

2-93 

2-97 

3.01 

3.06 

3.10 

3 -i 4 

3 -i 9 

3-24 

3-29 

3-34 

25 

66 

2.92 

2.96 

3.00 

3-04 

3.08 

3-13 

3 -i 7 

3.22 

3-27 

3 - 3 i 

3-37 

24 

67 

2.94 

2.98 

3.02 

3.06 

3.10 

3-15 

3.20 

3-24 

3-29 

3-34 

3-39 

23 

68 

2.96 

3.00 

3-04 

3-o8 

3 -i 3 

3 -i 7 

3.22 

3.26 

3 - 3 i 

3-36 

3-42 

22 

69 

2.98 

3.02 

3.06 

3.10 

3-15 

3 -i 9 

3-24 

3-29 

3-34 

3-39 

3-44 

21 

70 

3.00 

3-04 

3.08 

3.12 

3-17 

3.21 

3-26 

3-31 

3-36 

3 - 4 i 

3-46 

20 

7 i 

3.02 

3.06 

3.10 

3-14 

3-19 

3-24 

3.28 

3-33 

3-38 

3-43 

3-48 

19 

72 

3-04 

3.08 

3.12 

3.16 

3-21 

3-25 

3-30 

3-35 

3.40 

3-45 

3*50 

18 

73 

3-05 

3-09 

3-i4 

3.18 

3.22 

3-27 

3-32 

3-37 

3-42 

3 47 

3-52 

17 

74 

3 07 

3-n 

3 -i 5 

3.20 

3-24 

3-29 

3-33 

3-38 

3-44 

3 49 

3 54 

16 

75 

3.08 

3 -i 3 

3-17 

3.21 

3.26 

3 30 

3-35 

3-40 

3-45 

3-50 

3-56 

15 

76 

3.10 

3 -i 5 

3.1S 

3-23 

3.28 

3-32 

3-37 

3-42 

3-47 

3-53 

3-58 

14 

77 

3-n 

3-15 

3-19 

3-24 

3-29 

3-33 

3-38 

3-43 

3-48 

3-54 

3-59 

13 

78 

3.12 

3.16 

3.21 

3-25 

3-30 

3-34 

3-39 

3-44 

3-49 

3-55 

3.60 

12 

I 79 

3 -i 3 

3.18 

3.22 

3.26 

3 - 3 i 

3.36 

3 - 4 i 

3 46 

3 - 5 i 

3-56 

3.62 

IT 

80 

3-i4 

3-19 

3-23 

3-27 

3-32 

3-37 

3-42 

3-47 

3-52 

3-57 

3-63 

10 

! 81 

3 -i 5 

3.20 

3-24 

3.28 

3-33 

3-38 

3-43 

3-48 

3-53 

3-58 

3*64 

9 

82 

3.16 

3.20 

3-25 

3-29 

3-34 

3-39 

3-44 

3-49 

3-54 

3-59 

3-^5 

8 

1 8 3 

3-i7 

3.21 

3.26 

3-30 

3 35 

3-40 

3-45 

3-49 

3-55 

3.60 

3-66 

7 

! 84 

3.18 

3.22 

3.26 

3-31 

3-35 

3.40 

3-45 

3-50 

3-55 

3.61 

3.66 

6 

85 

3.18 

3.22 

3-27 

3 - 3 i 

3-3^ 

3 - 4 i 

3-4^ 

3 - 5 i 

3-56 

3.61 

3-67 

5 

1 86 

3-19 

3-23 

3-27 

3.32 

3-36 

3 - 4 i 

3-46 

3 - 5 1 

3-57 

3-62 

3-68 

4 

87 

3-i9 

3-23 

3.28 

3-32 

3-37 

3-42 

3 47 

3.52 

3-57 

3.62 

3.68 

3 

88 

3-i9 

3-23 

3.28 

3-32 

3-37 

3.42 

3 47 

3-52 

3-57 

3.62 

3-68 

2 

89 

3 -i 9 

3-24 

3.28 

3-33 

3-37 

3-42 

3-47 

3-52 

3-57 

3-63 

3.68 

1 

90 

3-19 

3-24 

3.28 

3-33 

3-37 

3-42 

3 47 

3 52 

3 57 

3-63 

3.68 

0 












































































77 o 


LONGITUDE , 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 
















































































A, B, C STAB FACTORS. 


77 1 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 


i 

742° 

74° 

75° 

75i° 

755° 

75s° 

46° 

2.69 

2-73 

2.78 

2.82 

2.87 

2.92 

47 

2.74 

2.78 

2.83 

2.87 

2.92 

2-97 

48 

2.78 

2.82 

2.87 

2.92 

2-97 

3.02 

49 

2 82 

2.87 

2.92 

2.96 

3.01 

3-07 

50 

2.87 

2.91 

2.96 

3.01 

3.06 

3 -n 

5 i 

2.9I 

2.95 

3.00 

3-05 

3.10 

3.16 

52 

2.95 

3.00 

3-04 

3-09 

3 -i 5 

3.20 

53 

2-99 

3.04 

3-09 

3-14 

3-19 

3.24 

54 

3-03 

3-oS 

3-13 

3 -i 8 

3-23 

3-29 

55 

3-07 

3 *ii 

3-16 

3.22 

3-27 

3-33 

56 

3.10 

3-15 

320 

3.26 

3 - 3 i 

3-37 

57 

3-14 

3.19 

3-24 

3-29 

3-35 

3 - 4 i 

58 

3-17 

3.22 

3.28 

3-33 

3-39 

3-45 

59 

3.21 

3.26 

3 - 3 i 

3-37 

3.42 

3-48 

60 

3-24 

3-29 

3-35 

3 - 4 ° 

3 - 4 ^ 

3-52 

61 

3-27 

3-33 

3-38 

3-44 

3-49 

3-55 

62 

3 - 3 ° 

3.36 

3-41 

3-47 

3-53 

3-59 

63 

3-33 

3-39 

3-44 

3-50 

3-56 

3.62 

64 

3-36 

342 

3-47 

3-53 

3-59 

3 - 65 , 

65 

3-39 

3-45 

3-50 

3-56 

3.62 

3.68 

66 

3-42 

3-47 

3-53 

3-59 

3-65 

3 - 7 i 

67 

3-44 

3-50 

3-56 

3.62 

3-68 

3-74 

68 

3-47 

3-53 

3.58 

3.64 

3-70 

3-77 

69 

3-49 

3-55 

3.61 

3-67 

3-73 

3-79 

70 

3-52 

3-57 

3-^3 

3-69 

3-75 

3.82 

7 i 

3-54 

3.60 

3*65 

3 - 7 i 

3-78 

3-84 

72 

3-56 

3.62 

3-67 

3-74 

3.80 

3.86 

73 

3-58 

3 64 

3*69 

3-76 

3.82 

3-89 

74 

3.60 

3-65 

3 - 7 i 

3-78 

3-84 

3 - 9 i 

75 

3.61 

1 3-67 

3-73 

3-79 

3.86 

3-92 

76 

3-64 

3-69 

3 75 

3.82 

3-88 

3-94 

77 

3-65 

3-70 

3-76 

3-83 

3-89 

3 - 9 ^ 

78 

3.66 

j 3-72 

3-78 

3-84 

3 - 9 i 

3-97 

79 

3-67 

3-73 

3-79 

3.86 

3-92 

3-99 

80 

3.68 

3-74 

3 . 8 i 

3.87 

3-93 

4.00 

81 

3-70 

3-75 

3.82 

3-88 

3-94 

4.01 

82 

3 7 i 

3-76 

3-83 

3-89 

3 - 9 6 

4.02 

83 

3-72 

3-77 

3*84 

3 - 9 ° 

3-96 

4-03 

84 

3 72 

3-78 

3-84 

3-91 

3-97 

4.04 

85 

3-73 

3-79 

3.85 

3-91 

3-98 

4-05 

86 

3-73 

3-79 

3-85 

3-92 

3 - 9 8 

4-05 

87 

3 - 74 

3-79 

1 3-86 

3-92 

3-99 

4.06 

88 

3-74 

3.80 

^.86 

3-92 

3-99 

4.06 

89 

3-74 

3.80 

3.86 

3-93 

3-99 

4.06 

90 

3-74 

3.80 

3.86 

3-93 

3 - 99 | 4 -oo 


7 6° 

76T 

76* 0 

7 6|° 

77 ° 

77 *° 


2.97 

3.03 

3.08 

3-14 

3.20 

3.26 

44 ° 

3.02 

3.08 

3 -i 3 

3-19 

3-25 

3 - 3 i 

43 

3 07 

3-13 

3 *i 8 

3-24 

3-30 

3-37 

42 

3.12 

3.18 

3-23 

3-29 

3-35 

3-42 

4 i 

3 -i 7 

3-22 

3.28 

3-34 

3 - 4 i 

3-47 

40 

3 21 

3-27 

3-33 

3-39 

3-45 

3-52 

39 

3.26 

3 - 3 i 

3-38 

3-44 

3.50 

3-57 

38 

3-30 

3-36 

3-42 

3-48 

3-55 

3.62 

37 

3-34 

3-40 

3-47 

3-53 

3.60 

3-67 

36 

3-39 

3-45 

3 * 5 i 

3-57 

3-64 

3 - 7 i 

35 

3-43 

3-49 

3 - 55 ! 

3.62 

3.68 

3-76 

34 

3-47 

3-53 

3-59 

3.661 

3-73 

3 -H 

33 

3-51 

3-57 

3-63! 

3.70 

3-77 

3-84 

32 

3-54 

3.61 

3.67 

3-74 

3.81 

3.88; 

3 i 

3-58 

3*64 

3 - 7 i 

3.78 

3-85 

3-92 

30 

3.62 

3-68 

3-75 

3.82 

3-89 

3-96 

29 

3-65 

3-72 

3-78 

3 - 8 5 

3-92 

4.00 

28 

3.68 

3-75 

3.82 

3-89 

3-96 

4.04 

27 

3-72 

3-78 

3-85 

3-92 

4.00 

4.07 

26 

3-75 

3.81 

3-88 

3-95 

4-03 

4.11 

25 

3-78 

3-84 

3 - 9 i 

3-99 

4.06 

4.14 

24 

3.81 

3-87 

3-94 

4.02 

4.09 

4.17 

23 

3-83 

3-90 

3-97 

4-05 

4.12 

4.20 

22 

3.86 

3-93 

4.00 

4.07 

4*15 

4.23 

21 

3-89 

3-95 

4-03 

4.10 

4.18 

4-25 

20 

3 - 9 i 

3-98 

4 05 

4-13 

4.20 

4.28 

19 

3-93 

4.00 

4.07 

4-15 

4-23 

4 . 3 i 

18 

3-95 

4.02 

4. TO 

4 17 

4-25 

4-33 

17 

3-97 

4.04 

4.12 

4.19 

4.27 

4-36 

16 

3-99 

4.06 

4.14 

4.21 

4.29 

4-38 

15 

4.01 

4.08 

4.16 

4.23 

4 - 3 i 

4.40 

14 

4 -03 

4.10 

4.17 

4.25 

4-33 

4.41 

13 

4.04 

4.11 

4.19 

4.27 

4-35 

4-43 

12 

4.06 

4.13 

4.21 

4.28 

4-36 

4-45 

11 

4.07 

4.14 

4.22 

4 30 

4-38 

4.46 

10 

4.08 

4.16 

4-23 

4-31 

4-39 

4.48 

9 

4.09 

4.17 

4.24 

4-32 

4.40 

4.49 

8 

4.10 

4.18 

4-25 

4-33 

4.41 

4-50 

7 

4.11 

4.18 

4.26 

4-34 

4.42 

4 - 5 i 

6 

4.12 

4.19 

4.27 

4-35 

4-43 

4 - 5 i 

5 

4.12 

4.20 

4.27 

4-35 

4-43 

4-52 

4 

4-13 

4.20 

4.28 

4 - 3 ^ 

4-44 

4-53 

3 

4.13 

4.20 

4.28 

4-36 

4.44 

4-53 

2 

413 

4.21 

4.28 

4.36 

4.44 

4-53 

1 

4-13 

4.21 

4.28 

4-36 

4.44 

4-53 

0 




































































































772 


LONGITUDE. 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 



772 ° 

773 ° 

78° 

7 8i° 


0 

e** 

00 

— 

79 ° 

79 i° 

795 ° 

79 l° 

|l 8 °° 

-1 

< 

1° 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

. IO 

1 

. 10 

. 10 

89 

2 

. l6 

. 16 

•17 

•17 

.18 

. 18 

.18 

.19 

.19 

.20 

.20 

88 

3 

.24 

• 25 

•25 

. 26 

.26 

•27 

•27 

. 28 

. 29 

• 29 

•30 

87 

4 

•32 

•33 

•34 

•34 

•35 

•36 

•37 

•37 

•38 

•39 

.40 

86 

5 

.40 

.41 

.42 

•43 

•44 

•45 

.46 

•47 

. 48 

•49 

• 50 

85 

6 

•49 

•49 

• 5 i 

• 5 i 

•52 

•54 

•55 

•56 

•57 

•59 

. 60 

84 

7 

• 56 

•57 

•59 

. 60 

.61 

. 62 

.64 

.65 

.67 

.69 

.70 

83 

8 

.64 

.66 

.67 

.68 

.70 

• 7 i 

•73 

•75 

.76 

•78 

. 80 

82 

9 

.72 

•74 

•75 

• 77 

•78 

. 80 

. 82 

.84 

.86 

.88 

.90 

81 

10 

. 80 

. 82 

.84 

•85 

•87 

.89 

9 i 

•93 

•95 

.98 

1.00 

80 

ii 

.88 

.90 

.92 

•94 

. 96 

.98 

1.00 

1.02 

1.05 

1.07 

1.10 

79 

12 

.96 

.98 

1.00 

1.02 

1.04 

1.07 

1.09 

1.11 

1.14 

1 • 17 

1.20 

78 

13 

1.04 

1.06 

1.08 

1.10 

1 • I 3 

115 

1.18 

1.21 

1.23 

1.26 

1.30 

77 

14 

1.12 

1.14 

1.16 

1.19 

1.21 

1.24 

1.27 

1.30 

i -33 

1.36 

1-39 

76 

15 

1.20 

1.22 

1.25 

1 .27 

1.30 

i -33 

1.36 

1-39 

1.42 

1.46 

1.49 

75 

16 

1.28 

1.30 

1-33 

i -35 

1.38 

1.41 

1.44 

1.48 

1 • 5i 

1-55 

i -59 

74 

17 

i -35 

1.38 

1.40 

1.44 

1.47 

1.50 

i -53 

i -57 

1.60 

1.64 

1.68 

73 

18 

i -43 

1.46 

1.49 

1.52 

i -55 

1.58 

1.62 

1.66 

1.70 

1.74 

1.78 

72 

19 

1.51 

1-53 

i -57 

1.60 

1.63 

1.67 

1.71 

1-75 

1.79 

1.83 

1.87 

7 i 

20 

1.58 

1.61 

1.65 

1.68 

1.72 

1 / j 

1.79 

1.83 

1.88 

1.92 

1.97 

70 

21 

1-65 

1.69 

1.72 

1.76 

1.80 

1.84 

1.88 

1.92 

1.97 

2.01 

2.06 

69 

22 

i -73 

1-77 

1.80 

1.84 

1.88 

1.92 

1.96 

2.01 

2.06 

2.11 

2.16 

68 

23 

1.81 

1.84 

1.88 

1.92 

1.96 

2.00 

2.05 

2.09 

2.14 

2.20 

2.25 

67 

24 

1.88 

1.92 

1.96 

2.00 

2.04 

2.08 

2.13 

2.18 

2. 23 

2.29 

2-34 

66 

25 

i -95 

1.99 

2.03 

2.07 

2.12 

2.17 

2.22 

2.27 

2.32 

2.38 

2-43 

65 

26 

2.02 

2.07 

2.11 

2.15 

2.20 

2.25 

2.30 

2-35 

2.41 

2.46 

2.52 

64 

27 

2.10 

2.14 

2.18 

2.23 

2.28 

2-33 

2.38 

2-43 

2.49 

2-55 

2.61 

63 

28 

2.17 

2.21 

2.26 

2.31 

2.36 

2.41 

2.46 

2.52 

2. 58 

2.64 

2.70 

62 

29 

2.24 

2.28 

2-33 

2.38 

2-43 

2.48 

2-54 

2.60 

2.66 

2-73 

2.79 

61 

30 

2.31 

2.36 

2.40 

2.46 

2.51 

2.56 

2.62 

2.68 

2-74 

2.81 

2.88 

60 

3 i 

2.38 

2-43 

2.48 

2-53 

2. 58 

2.64 

2.70 

2.76 

2.83 

2.89 

2.97 

59 

32 

2-45 

2.50 

2-55 

2.60 

1 

2.66 

2.72 

2.78 

2.84 

2.91 

2.98 | 

3-05 

58 

33 

2.52 

2-57 

2.62 

2.67 

2-73 

2.79 

2.85 

2.92 

2.99 

3.06 

3 -i 4 

57 

34 

2.58 

2.64 

2.69 

2-75 

2.80 

2.87 

2-93 

3.00 1 

3-07 

3-14 ! 

3.22 

56 

35 

2.65 

2.70 

2.76 

2.82 

2.88 

2-94 

3.01 

3.08 J 

3 15 

3.23 

3-30 

55 

36 

2.72 

2.77 

2.83 

2.89 j 

2-95 

3.01 

3.08 

3-15 

! 

3-23 

3-30 

3-38 

54 

37 

2. 78 

2.84 

2.90 

2-95 

3.02 

3 -°8 

3-15 

3-23 

3-30 

3-38 

3-47 

53 

38 

2.85 

2.90 

2 .()(» 

3.02 1 

3-09 

3-i6 

3-23 

3-30 1 

3-38 

3-46 

3-55 

52 

39 

2.91 

2.97 

3-03 

3 -og 

3.16 

3-23 

3-30 

3-37 1 

3-45 

3-53 

3.62 

5 i 

40 

2-97 

3-03 

3 -og 

3.16 

3.22 

3-29 

3-37 

3-45 

3-53 

3 -6i 

3-70 

50 

4 i 

3-03 

3 09 

3-i6 

3.22 

3-29 

3-36 

3-44 

3-52 

3.60 

3-69 

3-78 

49 

42 

3 -og 

3 -i 5 

3.22 

3 29 

3-36 

3-43 

3 . 5 i 

3-59 

3-67 

3-76 

3-85 

48 

43 

3 -i 5 

3-21 

3.28 

3-35 

3-42 

3-50 

3-57 

3.66 

3-74 

3-83 

3-93 

47 

44 

3-21 

3-27 

3-34 

3-41 

3-48 

3-56 

3-64 

3-72 

3.81 

3 - 9 i ! 

4.00 

46 

45 

327 

3-33 

3 -40 | 

3-47 

3-55 j 

3.62 

3 - 7 i 

3-79 

3-88 

3-97 

4.07 

45 

-1 1 







































































A, B , C STAR FACTORS. 


773 


Table LXX. 

FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS. 



0 

r-«|<N 

77 3° 

78° 

CO 

0 

0 

00 

r>. 

1 

O 

00 

79° 

0 

ON 

792 ° 

79l° 

8o° 

i 

46° 

3-32 

3-39 

3-46 

3-53 

3.61 

3-69 

3-77 

3.86 

3-95 

4.O4 

4.I4 

44° 

47 

3-38 

3-45 

3-52 

3-59 

3-67 

3-75 

3-83 

3-92 

4.01 

4.II 

4.21 

43 

48 

3-43 

3.50 

3-57 

3-65 

3-73 

3.81 

3-89 

3-98 

4.08 

4.18 

4.28 

42 

49 

3-49 1 

3-56 

3- 6 3 

3-7i 

3-79 

3-87 

3.96 

4-05 

4.14 

4.24 

4-35 

4i 

50 

3-54 

3-6i 

3-68 

3-76 

3-84 

3-93 

4.02 

4.11 

4.20 

4-30 

4.41 

40 

5 i 

3-59 

3-66 

3-74 

3.82 

3-90 

3-98 

4.07 

4.17 

4 26 

4-37 

4.48 

39 

52 

3*^4 

3-7i 

3-79 

3 87 

3-95 

4.04 

4-13 

4.22 

4-32 

4-43 

4-54 

38 

53 

3-69 

3-77 

3-84 

3-92 

4 01 

4.09 

4.19 

4.28 

4-38 

4.49 

4.60 

37 

54 

3-74 

3.81 

3-89 

3-97 

4.06 

4-i5 

4.24 

4-34 

4 44 

4-55 

4.66 

36 

55 

3-78 

3.86 

3-94 

4.02 

4.11 

4.20 

4.29 

4-39 

4.50 

4.60 

4.72 

35 

56 

3-83 

3-91 

3-99 

4.07 

4.16 

4-25 

4-34 

4.44 

4-55 

4.66 

4-77 

34 

57 

3-88 

3-95 

4.04 

412 

4.21 

4-30 

4-39 

4-50 

4.60 

4.72 

4-83 

33 

58 

3.92 

4.00 

4.08 

4.16 

4-25 

4-35 

4-44 

4-55 

4-65 

4-77 

4.88 

32 

59 

3-9^ 

4.04 

4.12 

4.21 

4-30 

4-39 

4.49 

4.60 

4.70 

4.82 

4.94 

3i 

60 

4.00 

4.08 

4.17 

4-25 

4-34 

4*44 

4-54 

4.64 

4-75 

4.87 

4-99 

30 

61 

4.04 

4.12 

4.21 

4.29 

4-39 

4.48 

4-58 

4.69 

4.80 

4.92 

5-04 

29 

62 

4.08 

4.16 

4-25 

4-34 

4-43 

4-53 

463 

4-73 

4-85 

4.96 

5.08 

28 

63 

4.12 

4.20 

4.29 

4-38 

4-47 

4*57 

4.67 

4.78 

4.89 

5-oi 

5-13 

27 

64 

4-15 

4.24 

4-32 

4 4i 

4 5i 

4.61 

4.71 

4.82 

4-93 

5-05 

5.18 

26 

65 

4.19 

4.27 

4-36 

4-45 

4 55 

4-65 

4-75 

4.86 

4-97 

5-09 

5-22 

25 

66 

4.22 

4-3i 

4.40 

4-49 

4-58 

4.68 

4-79 

4.90 

5-oi 

5.14 

5-26 

24 

67 

4.26 

4-34 

4-43 

4-52 

4.62 

4.72 

4.82 

4.94 

5-°5 

5.18 

•5-30 

23 

68 

4.28 

4-37 

4.46 

4-55 

4-65 

4-75 

4.86 

4-97 

5-09 

5-21 

5.34 

22 

69 

4-32 

4.40 

4.49 

4-58 

4.68 

4-79 

4.89 

5.00 

5-12 

5*25 

5.38 

21 

70 

4-34 

4-43 

4-52 

4.61 

4.71 

4.82 

4-93 

5-04 

5.16 

5-28 

5-4i 

20 

7i 

4-37 

4.46 

4-55 

4.64 

4-74 

4.85 

4.96 

5-07 

5-19 

5-32 

5-45 

19 

72 

4-39 

4.4S 

4-57 

4.67 

4-77 

4.88 

4.98 

5.10 

5.22 

5-34 

5-48 

18 

73 

4.42 

4-51 

4.60 

4.70 

4.80 

4.90 

501 

5.i3 

5.25 

5-37 

5-5i 

17 

74 

4.44 

4-53 

4.62 

4.72 

4.82 

4-93 

5-04 

5.15 

5 27 

5-40 

5-53 

16 

75 

4.46 

4-55 

4-65 

4-74 

4.84 

4-95 

5.06 

5.18 

5-30 

5-43 

5.56 

15 

76 

4.48 

4-57 

4 67 

4.76 

4.87 

4-97 

5.09 

5.20 

5-32 

5-45 

5.59 

14 

77 

4-50 

4-59 

4.68 

4.78 

4.89 

4.99 

5-11 

5.22 

5-35 

5-47 

5.61 

13 

78 

4-52 

4.61 

4.70 

4.80 

4.91 

5.01 

5.13 

5-24 

5-37 

5-50 

5-63 

12 

79 

4-54 

4-63 

4.72 

4.82 

4.92 

5.03 

5-i4 

5-26 

5-39 

5-52 

5.65 

11 

80 

4-55 

4.64 

4-74 

4.84 

4.94 

5.05 

5-i6 

5.28 

5-40 

5-54 

5-67 

10 

81 

4-56 

4-65 

4-75 

4-85 

4-95 

5.06 

5.18 

5-30 

5-42 

5.55 

5-69 

9 

82 

4-57 

4 67 

4.76 

4 86 

4-97 

5.08 

5-i9 

5.3i 

5-43 

5-56 

5.70 

8 

83 

4-59 

4.6S 

4.78 

4.87 

4.98 

5-09 

5.20 

5-32 

5-45 

5-58 

5.72 

7 

84 

4.60 

4.69 

4-79 

4.88 

4.99 

5.10 

5-21 

5-33 

5-46 

5-59 

5-73 

6 

85 

4.60 

4.69 

4-79 

4.89 

5.00 

5-11 

5.22 

5-34 

5.47 

5.60 

5-74 

5 

86 

4.61 

4.70 

4.80 

4.90 

5.00 

5.11 

5-23 

5-35 

5-47 

5.6i 

5-74 

4 

87 

4.62 

4-71 

4.81 

4.90 

5-Oi 

5-i2 

5.23 

5-35 

5.48 

5.61 

5.75 

3 

88 

89 

4.62 

4.62 

4-7i 

1 4.71 

4.81 

4.81 

4.91 

4.91 

5-01 

5.01 

5-i2 

5.12 

5-24 

5-24 

5-36 

5-36* 

5.48 

5.49 

5.61 

5.62 

5-75 

5-76 

2 

1 

90 

4.62 

» 

j 4-71 

j 4'8i 

4.91 

5.02 

5-i3 

5.24 

5.36 

5.49 

5.62 

5.76 

0 















































































774 


LONGITUDE. 


335. Comparison of Time.—After time has been thus 
observed the chronometers at the two stations should be 
compared by telegraph. This constitutes the automatic 
exchange of signals. The chronometer at one station being 
in circuit with the chronograph and recording upon it, that 
at the other station is switched into the telegraphic circuit, 
by which it is brought to the first station and switched 
into the local circuit there, so that the two chronometers 
register upon the same chronograph, their beats being marked 
side by side by the same pen. After this has gone on for a 
minute or more the operation is reversed, the chronometer at 
the first station is switched into the telegraphic circuit and 
made to record upon the chronograph with the chronometer 
at the second station. Of course the observers are informed 
of the hour and minute at which the joint record upon the 
several chronographs begins. 

The arbitrary exchange of signals is made as follows: Each 
chronometer recording on its own chronograph as usual, and 
each local circuit being connected with the main-line circuit 
through a relay, the observer at one station breaks the circuit 
by means of the main-line talking-key, which break is recorded 
on the chronograph sheets at both stations. The breaks are 
repeated at every two seconds for at least one full minute. 
The operation is then reversed by the observer at the second 
station, making the breaks which are recorded at both stations 
as before. The differences of time between the chronometers 
at the two stations are read from the chronograph sheets at 
each station and corrected for error of the chronometers. 
The results from the two chronograph sheets will differ by an 
amount equal to twice the time occupied in transmission of 
signals. The mean of the two is therefore the approximate 
difference of longitude. 

This result is yet to be corrected for personal equation , 
or the difference between the errors of observing of the two 


COM TAR ISON OF TIME. 


775 


observers. Every observer has the habit of recording a 
transit a little too early or too late, the difference between two 
observers not infrequently being as great as a fourth of a 
second. To measure this difference, the observers usually 
meet, preferably at the known station, both before and after 
the campaign, and observe for time each with his own in¬ 
strument, or with one similar in all respects to that used in 
the campaign. A comparison of the time determinations 
made by the two observers gives an approximation to the 
personal equation. 

A better method, but one not always practicable, is for 
the observers, having completed half of the observations 
for time and longitude, to exchange stations for the re¬ 
mainder of the work. The mean of the results before and 
after exchange of stations will practically eliminate personal 
equation. 

There is one error incident to this work which cannot be 
eliminated. This is the unequal attraction of gravity, or 
local attraction , or, as it is sometimes called, station error. 
The neighborhood of a mountain mass will attract the plumb- 
line and deflect the spirit-level to such an extent as to cause 
serious errors in astronomical determinations of latitude and 
time. The same result is frequently produced by a difference 
in density of the underlying strata of rock, so that station 
errors of magnitude often appear where they are not expected. 
Indeed, the station error cannot be predicted with any cer¬ 
tainty either as to amount or even direction. 

The only practical method of even partially eliminating 
this error is to select a number of stations for astronomic 
location, under conditions as widely diverse as possible, con¬ 
nect them by triangulation, and by this means reduce all these 
astronomical determinations to one point, thus obtaining for 
this point a number of astronomic determinations each hav¬ 
ing a different station error. The mean of these gives for 


LONGITUDE. 


// D 

this point a position from which—in part, at least—station 
error has been eliminated, and this mean position can be 
transferred back by means of the triangulation to the several 
astronomic stations, thus giving each of them a position 
similarly comparatively free from station error, a position so 
determined is referred to as a geodetic position. 


CHAPTER XXXVI. 




SEXTANT AND SOLAR ATTACHMENT. 

336. Sextant.—This is a hand instrument for measuring 
the angle subtended by any two objects. The principle of the 
measurement is dependent on the fact that the angle subtended 
by the eye by lines passing to it from two distant objects may 
be measured by so arranging two glasses that one object is 
looked at directly, while the image of the other is seen as re¬ 
flected from the silvered or mirrored surface of one glass to 
that of the other, and from the second to the eye. The 
mirror of the first glass is then moved so that the double re¬ 
flected image of the second object is made to coincide with 
the object as seen directly. 

The sextant is especially useful on exploratory surveys and 
at sea because of its lightness and portability and because it 
requires no fixed support. With it can be obtained results 
of sufficient accuracy for all the purposes of navigation 
and of exploratory determination of astronomic position. 
The sextant is also extensively used in measuring the heights 
of objects from the sea or from land, and in measuring horizon¬ 
tal angles between two objects, especially in hydrographic 
surveying for the location of soundings. 

Sextants are of various forms , which differ according to 
the maker. They are sometimes made of wood mounted with 
ivory, but such materials are liable to warp. The most satis¬ 
factory sextant for all-round surveying is made of brass with 

777 


7/8 


SEXTANT AND SOLAR ATTACHMENT. 


a silvered arc of sufficient extent to permit of measuring 
angles up to 8o°. 

The principal parts of the sextant (Fig. 183) are: 



Fig. 183. —Sextant. 


1. A mirror i, called the index glass, which is rigidly at¬ 
tached to the movable arm a, called the index arm; also 

2. A mirror h, called the horizon glass, rigidly attached to 
the frame of the instrument; and 

3. The arc on which the angles are read by means of the 
vernier at the end of the index arm. 

The planes of the two mirrors are so fixed as to be parallel, 
one to the other, when the vernier points to zero degrees. 

337. Adjustment of Sextant.—Among the more impor¬ 
tant adjustments of the sextant are those of— 

1. The index glass; 

2. The horizon glass; 

3. The telescope; 

4. Correction for index error. 

The reflecting surface of the index glass must be perpen¬ 
dicular to the plane of graduated arc of the instrument. To 
test it, set the index near the middle of the arc, then place the 
eye very near the index glass and plane of the instruments, 
































AD JU STM ENT OF SEXTANT. 


779 


and observe whether the reflected image of the arc forms a 
continuous or broken line with the arc as seen direct. If 
continuous, the glass is perpendicular to the plane of the in¬ 
strument. If the reflected image drops, the glass is leaning 
backward; if it rises, forward. The adjustment is made by 
means of a key on the back, the latter being turned to the 
left if the image is dropping, and to the right if rising. 

The reflecting surface of the horizon glass should be 
perpendicular to the plane of the instrument. To test this, 
put in the telescope and point it towards a star, holding the 
instrument vertical, then move the instrument until the re¬ 
flected image is in a horizontal line with the direct image. If 
it is exactly in coincidence with the direct image, the horizon 
glass and index glass must be parallel in that position, and as 
the index glass has been adjusted perpendicularly to that 
plane, in any position. If they do not coincide, put the ad¬ 
justing key on the screw at the back and turn to the right to 
move the reflected image to the right, and to the left to move 
it to the left. 

To make the line of collimation of the telescope parallel to 
the plane of the instrument , the sextant should be rested on a 
plane surface with the telescope directed at a well-defined 
point about 25 feet distant. Two objects of equal height are 
then placed on the extremities of the arc, and these serve to 
establish a plane of sight parallel to the arc. They may be 
two small sticks of sufficient height to make the plane of 
sight of the same height above the arc as is the line of collima¬ 
tion of the telescope. If the line of collimation now inter¬ 
sects the line defined by the two pointers, the instrument is in 
adjustment. If not, the error is corrected by the screws on 
the holder of the telescope. 

Another mode of performing this adjustment is the follow¬ 
ing : Place a telescope which has two wires in the field of view 
that are parallel to each other and equidistant from the center 
of the field, in the telescope ring of the sextant, and turn the 


780 


SEXTANT AND SOLAR ATTACHMENT.' 


eyepiece until the wires are parallel to the plane of the instru¬ 
ment. Measure the angular distance between the two objects 
which are apart as far, say, as 60 degrees or more; when the 
reflected and direct images are in contact on one wire, clamp 
the index firmly and make a precise contact by using the tan¬ 
gent screw; now move the instrument so as to bring the ob¬ 
ject on the other wire; if they remain in exact contact, the 
telescope is parallel to the plane of the instrument. If not, 
it may be adjusted by altering the screws in the ring so as to 
change the angle of the collar which holds the telescope. 

To correct the index error sight at some well-defined ob¬ 
ject, as a star, and move the index arm until the direct and 
reflected images coincide, when the vernier should read zero. 
If not, the difference maybe recorded as anindex error or be 
corrected by adjustment. 

338. Using the Sextant.—In measuring any angle with 
the sextant it is held in one hand in the plane of the two ob¬ 
jects. The telescope is then directed towards the fainter 
object by looking through the unsilvered portion of the 
horizon glass. With the other hand the index arm is then 
moved until the second object as seen by double reflection is 
brought in exact coincidence with that seen directly. 

If it is desired to read the horizontal angle between two 
objects which are at different elevations, some object, as a 
tree, building, or a plumb hung in line with one of them, must 
be found which is directly above or below it. The angle is 
then measured from this to the other object by holding the 
sextant horizontally in its plane. If no suitable object can 
be seen, some point about 60 degrees from one of the ob¬ 
jects may be selected and angles be read between each object 
and that point. The difference between these two angles 
will be approximately the horizontal angle. 

If it is desired to measure an angle between two objects 
which are very near together, the angle between each and a 
third object may be measured and the difference taken. 


SOLAR ATTACHMENT. 


781 


Should the angle to be read be too large to come within the 
range of one measurement of the arc, the sum of the angles 
between each object and an intermediate object may be 
measured. 

To measure vertical angles for determination of altitudes 
or for ascertaining heights of celestial bodies, the horizon is 
used as a reference point. At sea this is done by sighting 
the true horizon while on land an artificial horizon must be 
employed. The latter consists of a small bath of mercury 
protected from the wind by a glass cover. The observer stands 
or kneels near this reflecting surface of mercury and looks 
directly on the object the height of which is to be measured, 
and also at the reflection of this object in the mercury bath, 
and the contact is made between these two. The angle meas¬ 
ured from the reflecting surface is necessarily twice the an¬ 
gular elevation of the object observed above the true horizon. 

339. Solar Attachment.—The object of this instrument, 
which may be attached either to a compass or a transit, is 
the determining of the meridian , latitude , and time by ob¬ 
servation on the sun. It is extensively used in the sub¬ 
division of the public lands in the West. In the past the 
solar compass was mostly employed, but now little work is 
done with a compass, all meridians being run with the 
engineer’s transit by projecting from Polaris observations for 
azimuth, or with the solar attachment. 

This instrument was originally invented by Wm. A. Burt 
of Michigan, but at present there are several modifications of 
the original Burt attachment made by various manufacturers. 
There are, however, but two forms in popular use by sur¬ 
veyors ; these are the Burt Solar Attachment, as modified by 
Messrs. W. and L. E. Gurley, and the Smith Meridian At¬ 
tachment, made by Messrs. Young and Sons. The adjust¬ 
ment and use of these is described in the following Articles. 

340. Burt Solar Attachment.—This consists essentially 
of an axis which is parallel to the earth’s axis, and a line of 



7%2 


SEXTANT AND SOLAR ATTACHMENT. 


sight or pointer which is set at an angle to the instrumental 
polar axis equal to the declination of the sun for the time of 
observation. The polar axis is placed at right angles to the 
tube of the telescope by attaching it to the telescope tube by 
adjusting-screws. In a plane right angled to the polar axis 
is a small circle, graduated on its outer edge into fractional 
24 hours and called the equatorial or hour circle. Attached 
to the polar axis, and swinging about it with its lower side 
parallel to the plane of the hour axis, is an arm carrying a 
small arc with vernier attachment, called the declination arc. 

The polar axis is attached to the. telescope by a small 
circular disk of an inch and a half diameter, and on this as a 
pivot rests the enlarged base of the axis surrounded by the 
hour circle , the disk being attached to the base of the pivot 
of the polar axis by four capstan-headed screws which serve 
to adjust the polar axis. The hour circle can be fastened at 
any point desired by two flat-headed screws on its upper side, 
and the hours marked upon it are divided into 5 minutes of 
time, which are read by a small index fixed to the declination 
circle and moving with it. The declination arc is of about 5 
inches radius, divided into 30 degrees, reading by the vernier 
to single minutes. It is attached to the polar axis by a 
hollow cone or socket moving snugly upon it by a milled-head 
screw on top, and to this is securely fastened the declination 
arc by two large screws. The declination arc has two lenses 
and two silver plates on which equatorial and hour angles are 
ruled by parallel lines at right angles on two opposite ends 
of the radial arms of the sector; it has also a clamp and tangent 
movement, and the declination arc may be turned on its 
axis and one or the other of the solar lenses used, according 
as the sun is north or south of the equator. 

341. Adjustment of Burt Solar Attachment.—The ad¬ 
justments of a solar are simple. These are first to make the 
lines of collimation parallel to each other and at right angles 
to the polar axis, when the declination arc reads zero; and 


BUJiT SOLAR ATTACHMENT. 


783 



Fig. 184.— Graphic Illustration of the Solar Attachment. 




















































784 


SEXTANT AND SOLAR ATTACHMENT. 


second, to make the polar axis perpendicular to the telescope. 
In addition, the ordinary adjustments of the telescopic alidade 
must be made. 

The lines of collimation are made parallel by making each 
line parallel to the edges of the blocks containing them. This 
is done by removing the declination bar or bar carrying the 
lines of collimation, which is done by removing the clamp and 
tangent screws and the conical center with the small screws 
by which the arm is attached to the arc. Then a bar which 
is furnished with each instrument, called the adjusting-bar, is 
substituted for the declination arm ; and the conical center bar 
screwed into its place at one end and the clamp-screw into 
the other are inserted in the hole left by the removal of the 
tangent screw. The arm is then turned so as to bring the 
sun into one line of collimation, and then the bar is quickly 
revolved or turned over, but not end for end. If the image 
still falls in the square, the line of collimation is parallel to the 
two edges of the blocks. If not, the silver disk must be 
moved through half the apparent error of the sun’s image, and 
the same operation repeated. Then the bar must be reversed 
end for end by the opposite faces of the blocks upon it, and 
the other line of collimation adjusted until the image will re¬ 
main in the center of the equatorial lines. 

To adjust the polar axis the instrument is first carefully 
leveled, the tangent movement of the vertical arc of the tele¬ 
scope being used in connection with the leveling-screws of the 
striding-level of the alidade. Then the equatorial centers on 
top of the blocks are placed as closely together as practicable 
>vith obtaining a distinct view of a distant object. Having 
previously set the declination arm at zero, sight through the 
interval of the equatorial centers and blocks at some distant 
object, the declination arm being placed over either pair of 
t capstan-headed screws on the under side of the disk; now the 
instrument is turned on its axis and the same object sighted, 
while the declination arm is at the same time kept with one 


SMITH MERIDIAN ATTACHMENT. 


78 5 


hand upon the object originally sighted. If the sight line 
strikes either above or below, the instrument must be relev¬ 
eled by the two capstan-headed screws under the arm by such 
an amount as will eliminate half the error, and the operation 
again repeated until the sight strikes both objects in the same 
position of the instrument. The instrument may now be 
turned at right angles, keeping the sights still upon the same 
object as before, and if it does not strike the same point when 
sighted, the axis is not truly vertical in the second position 
of the instrument, and the correction must be made bv the 
capstan-headed screws under the declination arc by means of 
reversing it as before. 

To adjust the hour arc , which should read apparent time 
when the instrumen is set in the meridian, loosen the two flat¬ 
headed screws on top of the hour circle and with the hand 
turn the circle around until the index of the hour arc reads 
apparent time, when the screws may be fastened. 

342. Smith Meridian Attachment.—As this is a telescopic 
solar and thus permits of clearer definition of the sun and 
hence better work, it is preferred by many surveyors. This 
attachment is placed on the left side of the transit and is 
attached to the standard with a light plate by small butting- 
screws. A counterpoise is placed on the corresponding right 
side, and both can be easily removed when not in use. The 
solar telescope, C (Fig. 185), revolves in collars, and its line 
of collimation and axis of revolution coincide with the polar 
axis, PP. These collars are attached to the latitude arc , /, 
which has a horizontal axis, the whole being mounted on a 
frame which is attached to the transit standards, /, f. On 
the side of the telescope is fixed the declination arc , d , the 
vernier of which is attached to an arm, e, which turns on its 
axis a reflector, placed before the object-glass of the tele¬ 
scope. Both the latitude and declination arcs have tangent 
screws to impart slow motion. The arm holding the declina¬ 
tion vernier when placed at zero, is so arranged that the plane 


786 SEXTANT AND SO LAE ATTACHMENT. 

of the reflector makes an angle of 45 ° with the axis of the 
telescope. If the telescope is revolved on its polar axis, the 
reflected line of collimation will describe the celestial equator, 
thus by setting off any given declination north or south the 



N 

Fig. 185. —Smith Meridian Attachment. 


image of the sun may be kept in the field of view from its 
rising to its setting by revolving the telescope. The hour 
arc , h , is attached to the telescope and revolves at right angles 
with the polar axis. 

343- Adjustment of Smith Meridian Attachment.— 

These adjustments are made in the following manner and 
order, as arranged by Mr. Hargreaves Kippax: 

1. The adjustment of the line of collimation of the solar 
telescope is made first. It differs in no respect from the like 

































ADJUSTMENT OF SMITH MERIDIAN ATTACHMENT. 787 

adjustment in a Y level, and consists in rotating the telescope 
in its collars until the intersection of the center cross-hairs 
remains on a selected object or point throughout the possible 
extent of the rotation. In preparing for this adjustment, the 
apparatus may be revolved bodily on the axis of the latitude 
arc until the solar telescope becomes nearly level or assumes 
such other position as will conveniently observe the selected 
object or point. That sufficient light may be had, the re¬ 
flector should be placed edgewise; this can be conveniently 
done by removing the lug by which the vernier tangent-block 
is attached to the declination-arm. The tangent-block and 
the attached vernier-arm can then be swung so that the re¬ 
flector shall be edgewise, and it may be retained in that 
position by a rubber band or other device. Should this ex¬ 
pedient not give sufficient light, it will be necessary to remove 
the reflector and its attachments, not by removing the ring to 
which the bearings of the reflector are attached, but by care¬ 
fully removing the four screws which secure the two caps 
over the journals of the reflector-block. When replacing these 
caps, be careful not to screw them down so tightly as to cause 
too much friction on the journals, else there may be danger 
of disturbing the position ol the vernier-arm which is attached 
to the block by a small plug-screw, thus creating an index 
error unawares. It is well to have the object or point used 
in the adjustment of collimation somewhat remote, to avoid 
subsequent error through focusing on the sun. 

2. Having adjusted the collimation, it is convenient to see 
if the solar telescope travels in vertical plane. This can be 
best done while the reflector is removed. Carefully adjust 
the plate-levels of the instrument before attempting this 
adjustment. The angle of a building known to be vertical 
may be used, or, better still, an elevated object and its reflec¬ 
tion from an artificial horizon of mercury or other suitable 
fluid. It will be found convenient to remove the latitude 
clamp and tangent during this adjustment which is made by 


;88 


SEXTANT AND SOLAR ATTACHMENT. 


the four pairs of screws which attach the frame-plate to the 
standard. See test and adjustment 4, below. 

3. The next adjustment is to determine whether the Ime 
of collimation of the solar telescope is perpendicular to the axis 
of the latitude arc upon which the solar telescope revolves in 
altitude. This adjustment is supposed to be made perma¬ 
nently by the maker; the surveyor should not disturb it 
unless certain that a change is needed. Once perfected, the 
adjustment will seldom need attention. 

4. To test the parallelism of the tzvo telescopes , carefully 
adjust the transit telescope for collimation and elevate or 
depress it before making this test. A target may be used 
upon which two points are placed at a distance equal to the 
eccentricity of the solar telescope. By training the transit 
telescope on one point and the solar telescope on the other, 
the parallelism of the two lines of sight can be assured. 
A center mark may be placed on the first target, and the 
use of the second target be thus avoided. The adjustment 
is made by the four pairs of binding and butting screws by 
which the frame carrying the solar apparatus is attached 
to the standard of the transit. This adjustment may dis¬ 
arrange the second adjustment which has to be made by 
the same screws, and it may be found necessary to carry 
on both at the same time. After being satisfied of the 
preceding adjustments, the portions detached can then be 
replaced and attention given to the index errors of the lati¬ 
tude and declination arcs. 

5. To adjust index error of latitude arc , set at zero, clamp 
and place striding-level on telescope. Level with tangent 
screw. Reverse the level, and if the bubble returns to first 
position, the axis may be considered horizontal. If any de¬ 
viation is noticed, move the bubble half the distance by the 
tangent screw. Reverse the level, and if the bubble takes 
a like position in the opposite direction, the adjustment is 
accomplished. 


AZIMUTH AND LATITUDE WITH SOLAR. 789 

6. To adjust index error of declination arc , set off the lati¬ 
tude and observe the sun on the meridian by bringing its 
image exactly between the horizontal lines or equatorial 
wires. If any difference is noted between the observed and 
calculated declinations after correcting for refraction, move 
the arc by loosening the three screws on top of it until the 
difference is eliminated. Never attempt to remove the dec¬ 
lination index error by manipulating the small plug-screw 
which attaches the vernier-arm to the reflector. 

7. After adjusting as above, and making a careful ob¬ 
servation, it will usually happen that the transit telescope ivill 
still deviate one or two minutes from the meridian. If, on 
test, this appears to be a constant error, and you are satisfied 
with your former adjustments, the small deviation may be 
considered as the resultant of several residual undiscovered 
errors, and may be removed in the following manner. Ob¬ 
serve with the solar at 9 A.M. Turn transit telescope south, 
and note the error east or west of the meridian. Place transit 
telescope on the meridian with tangent screw. By means of 
the small butting-screws attaching the plate to the standard 
move the south end of the plate east or west, as was the 
error, until the image is precisely between the wires. Verify 
the adjustment by an observation at 3 P.M. 

8. To adjust equatorial wires , rotate the diaphram carry¬ 
ing the cross-wires by loosening the screws until the image 
follows the equatorial wires precisely. 

344. Determination of Azimuth and Latitude with Solar 
Attachment. —The declination of the sun is given in the 
American Ephemeris or Nautical Almanac and is calculated 
for apparent noon at Greenwich. It can also be determined 
from tables sold by makers of solar attachments. To deter¬ 
mine the declination for any other hour at a place in the 
United States, reference must be had to differences of time 
arising from longitude and the change of declination from 
day to day. The longitude of a place and therefore its 


790 


SEXTANT AND SOLAR ATTACHMENT. 


difference in time may be obtained merely from platting on 
a good map, or from a watch which is kept adjusted within a 
few minutes. The best time at which to use the solar attach¬ 
ment for the determining of a meridian is not at noon, when 
the sun is passing the meridian, nor early or late in the day, 
when refraction is greatest, but between 8 and 11 o’clock in 
the morning and 1.30 and 5 o’clock in the afternoon. 

The use of the solar attachment can best be explained by 
reference to an example. The following were prepared by 
Mr. A. F. Dunnington of the U. S. Geological Survey, from 
his field-notes: 

Example for Meridian. —Set the instrument over the 
corner of sections 7, 8, 17, and 18, T. 2 N., R. 5 E., of the 
Black Hills meridian, South Dakota. Level the transit and 
point the telescope approximately north with the aid of the 
magnetic needle. Knowing the latitude of the place, set the 
same off on the latitude arc. Having computed the declina¬ 
tion for the day and hour corrected for refraction, taken from 
the pocket Ephemeris, set it off on the declination arc. Place 
index at approximate local mean time on hour circle. Look 
into the solar telescope and the sun’s image should be seen in 
the field of view, but not between the equatorial wires. Now 
move the telescope of the transit into the meridian, and if the 
horizontal plates have been set at zero, any angle can be set 
off from the meridian and the course run. 

Record .—Aug. 4, 1898. Long. 103° 45'. At 7 h. 00 m. 
A.M., 1 . m. t., I set off 17 0 iL N. on the deck arc; 44 0 o8£' 
on the lat. arc, and determined a true meridian with the solar 
at the cor. of secs. 7, 8, 17, and 18, T. 2 N., R. 5 E., of the 
Black Hills meridian, South Dakota. 

Example for Latitude.— Some minutes before noon 
place instrument in position and level as before. Loosen 
clamp screw to the horizontal plate of the transit. Set off 
the computed declination for 12 M. corrected for refraction, 
and revolve the solar telescope in its collars until the index 


SOLAR ATTACHMENT TO TELESCOPIC ALIDADE. 7 91 


coincides with XII hours, making sure that this last setting is 
not disturbed. With the azimuth tangent screw of the transit 
bring the image in the field of view, and with the slow-motion 
screw of the latitude arc bring the image between the equato¬ 
rial lines. As the image leaves the wires repeat this opera¬ 
tion until the image appears to remain stationary for a few 
moments before leaving the wires in an opposite direction. 
At this moment the sun has reached its highest point, and 
the latitude of the place is read direct from its arc with the 
Smith meridian attachment, and the colatitude with the Burt 
attachment 

Record. —Aug. 31, 1897. Long. 103° 45'. At the cor. 
of secs. 13, 14, 23, and 24, T. 2 N., R. 3 E., of the Black 
Hills meridian, South Dakota, I set off 8° 22' N. on the 
deck arc; and at o h. o. 10 m. P.M., 1 . m. t., observe the sun 
on the meridian; the resulting latitude is 44 0 02N., which 
is about o'.2 1 less than the proper latitude. 

345. Solar Attachment to Telescopic Alidade. —As an 
instrument for use in topographic surveys the solar attachment 
has some advantages, especially in heavily timbered country, 
or where the magnetic declination is variable, as an aid to 
the rapid location of points in connection with the plane-table. 
Such locations are of necessity not of sufficient accuracy to 
permit of their being used in further extension of triangula¬ 
tion, but they are of sufficient accuracy ordinarily to permit 
of their being employed as tertiary control, either for the 
adjustment of traverses or the sketching-in of topographic 
details. 

The solar attachment to the telescopic alidade is used chiefly 
as a means of orienting the plane-table or placing it in true merid¬ 
ian when but one or two located points are visible; in other 
words, without the solar attachment a station to which sights 
have not yet been taken can be located only by means of the three- 
point problem (Art. 75) or reduction from three known stations. 


SEXTANT AND SOLAR ATTACHMENT. 


79 




With the aid of the solar attachment a resection location 
can be accurately made under most circumstances when two 
points only are in view. The method of procedure is as 
follows, and is rather similar to that employed in traversing 
with a plane-table when a magnetic needle is used for orienta¬ 
tion: Having set up the plane-table at the point the position 
of which is desired, the telescopic alidade is placed on the 
board with its fiducial edge parallel to a true north and south 
line which is ruled somewhere on the paper, and after the solar 
observation has been made the board is swung into true 
meridian and clamped. Now swing the far end of the alidade 
about the positions of first one and then the other of the two 
known points, and drawing lines along the edge of the ruler, 
the intersection of these lines is the position of the point 
occupied, and this position is made more than an approxima¬ 
tion by the third check secured by an intersection with a true 
meridian line obtained by the aid of the solar attachment. 

The unknown factor is the direction of the meridian. The 
latitude of the point may be determined by observation, as with 
the solar transit, but in the case of plane-table triangulation 
conducted on small scales and based on primary triangulation 
the latitude can be platted from the plane-table sheet with 
sufficient accuracy. This is then set off on the larger vertical 
arc of the telescopic alidade as a colatitude, so that the polar 
axis when in meridian may point to the pole. On the declina¬ 
tion arc is set off the declination for the time of the observa¬ 
tion. Now, with the plane-table leveled so that revolutions 
about its vertical axis may be in azimuth only, the board is 
revolved horizontally and with the line of sight about the polar 
axis until the image of the sun is brought between the equa¬ 
torial lines. Then the polar axis and the telescope will lie in 
meridian, and the instrument may be clamped and the meridian 
line ruled upon the board. 

345a. Traversing with Solar Alidade.—-Two forms of solar 
attachment to the Ul:s:opic alidade which have met with much 
favor and success are the Glenn Smith solar and the H. L. 



SOLAR ATTACHMENT TO TELESCOPIC ALIDADE. 792 a 

Baldwin solar, both devised by employees of the U. S. Geological 
Survey. The Glenn Smith solar is perhaps the more accurate, 
but more cumbersome of the two. It consists of an additional 
attachment to the tripod head whereby the latter may be ori¬ 
ented by a solar transit set upon it. Then the transit is slipped 
off the tripod spindle and a plane-table board substituted, 




Fig. i. 



Fig. 185a. —Baldwin Solar Alidade Attachment. 

which, by means of an index line on each part, falls into posi¬ 
tion and orientation. 

The Baldwin solar attachment does not involve the tripod 
head. The plane-table board remains at all times on the tripod, 
and to the telescopic alidade are attached an arc, circle and 
mirror which transform it into a solar. 

















































792 b SEXTANT AND SOLAR ATTACHMENT. 

With either attachment the object is to permit of rapid ori¬ 
entation of the plane-table board in true meridian, when fore- 
and back-sights are taken and distances plotted, as for compass 
plane-table traverse (Art. 81, p. 197), and the work progresses 
in like routine from one traverse station to the next. The result 
is much greater accuracy in the azimuths of the traverse survey. 

To use the Baldwin solar alidade place it over the tripod head 
on the plane-table with the leveling clamp slightly loose, and 
level by tapping gently on edge of board until bubbles are cen¬ 
tral. The plane-table should then be approximately oriented 
by use of the compass or otherwise, then clamped. Place the 
alidade on the north and south projection line nearest the 
platted location of the station, with the objective end south, 
bring the bubble of striding level to center and read the arc. 
From the reading subtract the latitude of place, the remainder 
will be the degrees and minutes of the latitude setting. 

Set off on the solar arc the declination of the sum corrected 
for longitude and refraction for the day and hour; for north 
declination the vernier will be toward the head of tangent 
screw, and for south declination toward the end of latter. Re¬ 
volve the telescope in its collar until the hour circle is set for 
local time, which will be standard time corrected for longitude 
and for equation of time (sun fast or slow). Now unclamp the 
plane-table and move carefully until the sun is in the field of 
view, preferably at center of field, and midway between the ex¬ 
treme stadia wires. The table will then be oriented and may 
be clamped, and sight lines be drawn to the next traverse station. 

If table can be nearly oriented by compass or otherwise, a 
slight horizontal movement will not injuriously affect the level¬ 
ing, but care in leveling at the beginning is necessary, as any 
change affects the azimuth, more especially if not level in an 
easterly and westerly direction, while any error in a northerly 
and southerly direction is eliminated when the first reading of 
the arc is taken, and therefore only the latitude setting is 
changed. These errors are greatly magnified near midday. 


CHAPTER XXXVII. 


PHOTOGRAPHIC LONGITUDES. 

346. Field-work of Observing Photographic Longi¬ 
tude.—A photographic camera of particularly stable and rigid 
form is set up, so that the image of the moon is about in the 
center of the plate, and a series of instantaneous exposures are 
made, allowing such an interval between the exposures that 
the moon’s images on the plate will not overlap; i.e., from 
i-J to 2% minutes, according to the moon’s age. After a set 
of, say, seven moon exposures the camera is left untouched 
until bright stars of approximately the same declination as 
the moon have arrived at the same point in the heavens. 
The camera is then opened for periods of 1 5 to 30 seconds, 
and the stars allowed to impress their trails on the plate. 
These star exposures should be repeated four or five times. 

It is obvious that if the local time of each moon and star 
exposure be known, such plate will give all data necessary to 
compute the moon’s position either in right ascension, declina¬ 
tion, azimuth, or lunar distance. 

In placing the sensitive plate in the slide, care must be 
taken that the pointed screws against which it rests are only 
allowed to puncture and not to scratch the gelatine film, 
otherwise the exact position of the plate will be uncertain. 
The camera should be set up so that the moon will, as nearly 
as can be judged, cross the center of the field in about seven 
or eight minutes, so that the resulting photograph will show 
its seven images distributed on each side of the center. As it 
is difficult to insure this, it is a good plan to make a larger 

793 


794 


PHOTOGRAPHIC LONGITUDES. 


number of moon exposures, that the measurer may select those 
most centrally situated. 

If the star exposures have been made first, it may be found 
that the moon does not cross the field near the center. In 
this case it will be necessary, after all the exposures have been 
completed, to move the camera, so that the moon is in the 
center, and take two or three additional exposures. These 
are only for the purpose of measuring the radius of the moon’s 
image on the plate. 

The moon exposures should be instantaneous. In making 
star exposures it is desirable to take two sets of stars in order 
to get trails both north and south of the moon. The essen¬ 
tial conditions are that the local time of each star and moon 
exposure should be known and that there should be a series 
of trails of at least two stars on the plate. To prevent possi¬ 
ble confusion it is advisable to make some small difference in 
the two sets of exposures by putting in one or two indicating 
exposures of twice the length of the others. 

The proper time for the commencement of the star ex¬ 
posures should be determined by calculation, not by looking 
through the sighting arrangement. After the camera has 
been clamped ready for the first exposure it should not be 
approached to a nearer distance than 3 or 4 feet until all the 
exposures are complete. The minimum magnitude of star 
that can be used with a clear sky is about a third-magnitude 
star. 

The following details of the process and the appended 
example have been worked out by Mr. Wm. J. Peters of the 
U. S. Geological Survey in connection with Capt.E.H.Hills’s, 
(R.E.) published description of his experiments. 

347. The Camera and its Adjustments.—The camera 
should be moderately heavy for good work, but may be 
lighter for approximate or exploratory work. Other things 
being equal, the longer the focal length the larger the scale of 
the photograph, and hence the more accurate the measure- 


THE CAMERA AND ITS ADJUSTMENTS. 


7 95 


ments. The means of transport available will probably be 
the guiding factor. The camera must be capable of being 
readily turned to any portion of the sky and of being firmly 
clamped in position. It is therefore best to use a photo-sur¬ 
veying camera or theodolite (Art. 125) or to mount the body 
between a pair of wyes with clamping arrangement in altitude, 
the wyes being on a base plate which can be rotated so that 
the whole instrument can be rotated. 

Provided the mounting be strong and stable, it can be of 
the roughest character, as it is not necessary to know any¬ 
thing whatever about the position of the camera at the mo¬ 
ment of exposure. The one essential is that the instrument 
shall not move during the whole time of exposure, often of sev¬ 
eral hours’ duration. The stand should be low, a height of 20 
inches to the base plate being ample. A tripod is probably 
best, provided it be firmly braced. The ends of the legs 
which rest on the ground should be flat, not pointed. The 
legs should be of well-seasoned wood, this being more con¬ 
stant under changes of temperature than metal. 

In designing the body of the camera two points must be 
borne in mind : first, that the focus, when once found, shall 
not require any alteration to compensate for change of tem¬ 
perature ; and secondly, that the center of the plate, i.e., 
the point where the axis of lens and camera cuts it, shall not 
shift. These conditions are perfectly fulfilled by making the 
body of stout brass tubing, which will expand or contract 
symmetrically, and the plate will therefore maintain the same 
position with reference to the axis of the instrument. 

It may be observed that it is of no importance to know 
the actual focal length of the lens , the quantity not being 
required in the formulas of reduction. Some form of finder , 
as a pin-hole and sight-vane, must be provided to enable the 
observer to direct the camera so that the center of the field 
falls at any desired point in the sky. The exposing arrange¬ 
ment that seems preferable is a simple flap shutter , actuated 


796 


PHO TOG PA PHIC L ONGJ TUBES. 


by a pneumatic ball, with sufficient length of tubing to allow 
the observer to keep at a distance of about four feet from 
the camera, and thus to enable it to maintain the high degree 
of stability essential to success. The slight shock of the 
opening of the light flap or blind does not move a heavy 
camera to any measurable extent. 

The plate-holder should be of metal, and must be provided 
with some means by which the position of the sensitive plate 
in the slide can be readily determined, and with an adjust¬ 
ment which will cause the plate to take up a position truly 
perpendicular to the optical axis. This is completely effected 
by making the plate rest against three sharp-pointed screws, 
which puncture the gelatine film and give three points from 
which the position of the plate can be exactly determined ; 
the necessary adjustment being made by moving the screws 
in or out. 

The lens employed must be one giving an absence of op¬ 
tical distortion over a large field, and must, therefore, be of 
the doublet form. A good lens of this class would show no 
appreciable distortion up to a distance of 7° from the center of 
the plate, which field is amply large for the purpose. 

The following definitions will aid in an understanding of 
the adjustments of the camera: 

1. The optical center of the plate is the point where the 
axis of the lens cuts it. 

2. The geometrical center of the plate is the foot of the 
perpendicular from the center of the lens to the plate. 

The adjustments required are: 

1. Focus; 

2. Finding the geometrical center of the plate; and 

3. Bringing the geometrical and optical centers into coin¬ 
cidence. 

Of these 1 and 2 are accomplished in one operation, 
as follows: The camera is placed in a vertical position, lens 
downwards, over a mercury bath. The plate-holder is in- 


MEASUREMENT OF THE PLATE. 


797 


serted and a piece of plate glass is placed, resting on three 
pivoted screws, in the position that a sensitive plate would 
occupy. The camera is then moved until the top surface, 
and therefore the bottom surface, of this glass plate is truly 
level. A mark is made at the middle of the lower surface of 
the plate, and by examining with an eyepiece the reflected 
image of this mark can be seen somewhere about the same 
position as the mark itself. The lens is moved in or out until 
the image is brought to a focus in the same plane as the ob¬ 
ject, and the glass plate or mark on it is moved until image 
and object coincide. The mark is then at the geometrical 
center of the plate and at the true focus of the lens. The 
center thus formed must coincide with the optical center, i.e., 
must lie over the axis of the lens. Should this be found not 
to be the case, the screws in the plate-holder must be altered 
until geometrical and optical centers coincide. A small error 
in this point will produce a quite appreciable error in the 
results, but should it be desired to test it there are well-known 
methods available which it is not necessary to describe here. 
It will, therefore, suffice if the geometrical center of the plate 
is brought to the axis of the camera. The points where the 
glass plate rests on the three screws, when adjustment is com¬ 
plete, are marked on the glass, and the latter then forms a 
gauge from which the center of any plate can be marked on it 
after exposure and development. It is obvious that this ad¬ 
justment is not one that is likely to be disturbed, and need, 
therefore, only be repeated at rare intervals. 

348. Measurement of the Plate.—The measurements 
must be made with a micrometer, but it is not necessary to 
• enter into a discussion of all the details of making such meas¬ 
urement of a photographic plate. The quantities to be meas¬ 
ured are the coordinates of each moon image and star trail 
from any two axes on the plate, expressed in any scale. The 
axes need not be mutually perpendicular, but they must be 
straight. A reseau may be used, this being a plate coated 


798 


PHOTOGRAPHIC LONGITUDES. 


with an opaque substance and ruled with two sets of fine 
transparent lines at intervals of 2 or 5 millimeters. It is 
placed in contact with the photographic plate previous to 
development, and, in that position, exposed to the light. 
The lines are, therefore, impressed on the plate and the 
reseau is developed together with the stars. The method is 
undoubtedly a very accurate one, but is not perhaps advisable, 
as adding another operation to be performed in the field. 
An alternative method is to use a positive reseau, i.e., black 
lines on a transparent ground, and to clamp the star plate 
and reseau plate, film to film, for measurement. A simple 
method, and one susceptible of quite sufficient accuracy, is to 
rule two axes on the gelatine film with a fi.ne needle. This 
has the advantage, in common with the first method, that 
the plate can be readily remeasured at any future time. 

The 7 neasurement of the coordinates of the star trails and 
moon’s bright limb call for no special remark; the only diffi¬ 
culty met with is when attempt is made to measure the moon’s 
radius in order to deduce the coordinate of the center. This 
is a point which has given a considerable amount of trouble, 
and calls for a somewhat detailed notice. The moon’s image 
on the plate is, in general, not circular, but is subject to two 
distortions , the first due to the difference of refraction on the 
upper and lower limbs, and the second due to the photo¬ 
graphic projection, the cone of rays from the moon through 
the center of the lens being cut obliquely by the plate. The 
moon image on the plate is therefore elliptical or very 
nearly so. 

The first distortion will vanish when the moon is at a 
sufficient altitude, and the second when the image is near the 
center of the plate. In this work a negligible error may be 
provisionally defined as a quantity less than one second of an 
arc. In this case the quantity sought is the radius of the 
image of the moon, and therefore a distortion up to 2 n in 
the diameter may occur, which means that the moon’s alti- 


MEASUREMENT OF THE PLATE. 


7 99 


tude atthe time of exposure must be at least 30°, and the 
distance of the image from the center of the plate must not 
exceed 2°. 

It is no doubt theoretically possible to devise a method 
for determining the position of the center of the moon’s image 
from measurements to the limb on the supposition that the 
latter is an ellipse. Practically this has not been found a 
success, nor is it necessary, providing one or two of the 
images are within the requisite 2° of the center, when they 
can be treated as circular and their radii easily measured. In 
the case of the other moon, more remote from the center of 
the plate, the distance from limb to center of image can be 
determined by a simple calculation. It is therefore required 
to have on the plate at least one moon image within 2° of the 
center. The observer will find no difficulty in fulfilling this 
condition, which, while greatly facilitating the measurement 
of the plate, is not absolutely essential to it. 

The problem of determining the radius of a central image 
is a comparatively simple one, and reduces itself to the ques¬ 
tion of finding the radius of a circle when a portion of the 
arc is given. It may at first sight be thought that, if the 
moon be more than half full, the diameter might be measured 
directly; but this is by no means the case, as will be apparent 
upon consideration. Except the moon be absolutely full the 
bright line will cover only a semicircle, and therefore, unless 
the measurement be made exactly on the line joining the ends- 
of the terminator, it will be erroneous. This line is quite 
impossible to select with any precision on the plate, especially 
if the moon be nearly full. 

One successful method is to rule a line across the image 
and measure the sine and versine. This is not very accurate 
for this reason : The cord cuts the limb obliquely, and conse¬ 
quently, in measuring its length in the micrometer, the cross¬ 
wire cuts the limb, while in measuring the versine the cross¬ 
wire is brought up, touching the limb. The edge of the image 


8 oo 


PHOTOGRAPH1C LONGITUDES. 


is not absolutely sharp, and the two measurements are not 
strictly comparable. 

A much superior method, in which all the contacts are 
symmetrical, is as follows : The microscope is fitted with two 

pairs of cross-wires inclined to each other at 
45 0 . The line of motion of the micrometer 
being along the wire OB, the moon image is 
made to touch OD and OC; the screw is then 
turned until EF touches the limb, and the 



B 


length OP is thus measured. The radius = —=-. 

T 2 - I 

It is to be noticed that the measurement of the radius is 
not necessarily an absolute one. Thus the observer may 
habitually make the wire encroach too much on the limb, and 
so measure the radius too small. This error, however, com¬ 
pletely disappears, since he will measure the distance from 
the coordinate axis to the limb with the same bias. Hence the 
coordinate of the limb will be increased exactly the same 
amount as the apparent radius is decreased, and the deduced 
coordinate of the center will not be affected. It is therefore 
important not only that the same observer should make both 
sets of measurements, which may be regarded as absolutely 
essential, but also that he should make them at the same time 
and under the same conditions of lighting. 

The two coordinate axes on the plate should be ruled 
-approximately parallel and perpendicular to the meridian 
through the center. If, for the final result, the moon’s right 
ascension is to be computed, which seems to be the preferable 
.method, it will be readily seen that much greater weight 
attaches to the coordinates perpendicular to the meridian than 
to those parallel to it. It is consequently desirable to make 
at least twice as many readings on the one set as on the 
other. The actual number of readings taken must be left to 
the taste of the individual observer; but for those who have 
no particular bias it is suggested that four readings of one 





COMPUTATION OF THE PLATE. 


801 


coordinate, eight of the other, and twenty-four for the radius 
of the image form a good working set. A plate with twelve 
star trails and seven moons can be measured thus in about 
four hours. 

349. Computation of the Plate.—The time of each ex¬ 
posure and the measured coordinates of moon and star images 
and of the center of the plate are all the data necessary for 
determining the position of the moon. The result can be 
expressed in any form, as right ascension, azimuth, or lunar 
distance, and the strictly correct course would therefore be to 
select whichever of these be changing most rapidly at the 
time, and reduce the result to that form. 

It may, however, be remarked: First, with regard to decli¬ 
nation, that the rate of change of this quantity is so variable, 
and will so often be too small to be of any value, that the 
extra labor spent in computing it will rarely be repaid. Sec¬ 
ondly, with regard to lunar distances, that this would only be 
applicable to certain stars; and in the case of a star near the 
same parallel of declination as the moon, the rate of change 
of the lunar distance is practically identical with the rate of 
change of the moon’s right ascension. If, therefore, right 
ascension be computed, then, to all intents and purposes, the 
lunar distance is also computed, and to give separate attention 
to the latter would be a waste of time. The best method to 
adopt as the standard one is the computation of the moon’s 
right ascension, which quantity is always changing at a practi¬ 
cally uniform rate, thus securing the same measure of precision 
in the results, whatever be the moon’s position. 

There is, in certain cases, some advantage in taking the 
computation of the moon’s azimuth, in that it disseminates 
one source of error, as will be seen later when the errors of 
the method are dealt with. The computation is, however, 
more laborious, and it seems doubtful whether it be in any 
case worth the additional labor. The only occasion where it 
would be at all desirable is when the moon is at a considerable 


802 


PHO TOGRA PHIC L ONGIT UD £S. 


distance from the meridian, and the observer is not near the 
equator. 

A lt P x — apparent right ascension (R. A.) and north-polar dis¬ 
tance (N. P. D.) of center of plate; 
a of p— apparent right ascension and north-polar distance of 
known star; 

a', p' — apparent right ascension and north-polar distance of 
unknown star or moon; 

a y p — true apparent right ascension and north-polar dis¬ 
tance of unknown star or moon; 
x y y = measured coordinates of star or moon; 

B, y r) = standard coordinates of star or moon (i.e., rectan¬ 
gular coordinates on a plane tangent to celestial 
spheres at AP perpendicular and parallel to the 
meridian through the center, expressed in parts of 
the radius); 

a, by Cy d, eyf — plate constants; 

6 = sidereal time (arc); 
t — hour-angle; 

6 = moon’s declination; 
n — moon’s horizontal parallax; 
p = earth’s radius; 

0' = reduced latitude; 
qy q 0 = auxiliary angles. 

To compute the moon’s right ascension from the plate the 
following is the procedure : 

(1) Assume any epoch for the plate, conveniently the 
sidereal time T of the first moon exposure. 

(2) The center of each star trail exposed at a sidereal time 
5 is to be regarded as an imaginary star whose north-polar 
distance equals that of the star, and whose right ascension 
= R. A. of star — (T — S). 

(3) Compute true L. S. T. of epoch (in arc). 

(4) Estimate by any approximate method A and P, the 
R. A. and N. P. D. of the center of the plate. 


COMPUTATION OF THE PLATE . 


803 


(5) Select three star trails, of which not more than two 
can be of the same star, and compute their £, // by formulas: 


£ _ tan (a % — A) sin q 0 > 
cos {P'— q 0 ) 


(168) 


V — tan {P — q o ) ; .(169) 

where tan ? 0 = tan /, cos (a 0 - A). . . . (170) 

(6) Calculate approximately the six plate constants, 
a , £, c y dy e } f, from the six equations thus furnished of the 
form 

0 

= ax — by — c ; 

V = ^ - ey —/; 


viz., one pair for each star. 

(7) With approximate values of the constants thus found, 
and the measured coordinates of the center of the plate, calcu¬ 
late the B,y rf of the center, and hence its corrected R. A. and 
N. P. D. form the approximate formulas. 

(8) With the new AP calculate £, r] of all the stars. 

(9) The plate constants can now be accurately determined 
from the comparison of the measured x, y, and the computed 
B,y rf for all the stars, either by least squares or by suitably 
grouping the stars in threes and taking the arithmetical mean 
of all the values thus found. This latter procedure is practi¬ 
cally as accurate and is much less laborious. 

(10) Calculate £, ?/ of all the moons and their a' from the 
formulas 

q = P — tan" 1 ;/,. 07 1 ) 


tan (a' — A) — 


£, cos (P — q) 
sin q 


(172) 


(11) With an assumed approximate longitude calculate the 
moon’s declination and horizontal parallax at time of each 
exposure, and deduce parallax in right ascension by ordinary 
formulas. 




804 


PHOTOGRAPHIC LONGITUDES. 


(12) There is thus obtained true right ascension of each 
image, and by adding the interval from assumed epoch we 
get moon’s true right ascension at each exposure, and hence 
Greenwich M. T. and longitude. 

350. Sources of Error.—The degree of accuracy of the 
longitude, as obtained from the photographic plate, is exhibited 
in Article 351 by examples taken at a place whose longitude 
is known. For their better comprehension it will be inter¬ 
esting to briefly discuss the errors that the method is liable 
to, their possible elimination, and their probable amount. 

The possible sources of error may be classified as follows: 

(a) Differential refraction ; 

(b) Aberration; 

(V) Flexures of camera; 

(J) Want of stability of camera; 

(e) Optical distortion of lens; 

(/) Lag in photographic action of a faint star; 

(g) Error in estimating position of center of plate; 

(//) Errors of measurement in the micrometer; 

(t) Clock errors (i.e., local time and clock rate); 

(j) Personal equation in making exposures; 

(k) Movement of moon during exposure; 

(/) Change of refraction between moon and star exposures. 

(a), (b), (c) will entirely disappear owing to differential 
nature of measurements between moon and stars. 

(< d) The necessity of a high degree of stability in the 
instrument has already been insisted upon, and nothing more 
need be said on the point. 

(e) With a suitable lens this source of error, which has 
been mentioned above, is quite negligible. 

(/) A faint star will not act on the plate as rapidly as a 
bright one, but there it no theoretical reason why this should 
cause any errors, as the measurements are taken to both ends 
of the star trails. 


SOURCES OF ERROR. 


805 

(<£*) 1 his is unlikely to have an appreciable effect on the 
result. Should the estimated position of the center differ by 
- fiom the true one, the position of the unknown star or 
moon would be wrong by a maximum of about 1". The 
resulting error of position varies directly as the error of the 
center, and as- the square of the distance of the star from the 
center. There should be no difficulty in finding the position 
of the center within 2'. 

(//) 1 he error in the micrometer may be due to: 

(1) Imperfections in the screws or scale; 

(2) Coordinate axes not being straight; 

(3) Errors of bisection on the image; 

(4) Errors due to the moon’s radius not being accurately 
measured ; 

(5) Distortion of the photographic film. 

Of these (1) can be eliminated by well-known methods, 
and need not be more than mentioned here; (2) should not 
amount to a measurable quantity; (3) will totally disappear, 
in so far as systematic errors of bisection are concerned, if the 
plate be reversed during the measurement; (4) has already 
been discussed; (5) has been proved negligible in the case 
of the Astrographic Chart plates. 

As an illustration of the degree of concordance that may 
be expected in a series of micrometer measurements of a 
moon or star image, the following set, which has been se¬ 
lected quite at haphazard, will be of interest. They are the 
measurements of one coordinate of the moon’s limb expressed 
in millimeters. As the length of the lens was 19.5 inches, a 
unit in the third decimal place represents very nearly o".5. 


Moon 


Mean. 

I 

53-024 

3 i 

36 

28 

32 

3 i 

33 

21 

53-030 

2 

49.067 

70 

60 

71 

72 

7 i 

76 

61 

49.070 

3 

45-096 

92 

01 

95 

00 

94 

03 

92 

45-097 

4 

41.130 

35 

41 

35 

38 

3 i 

4 i 

28 

41-135 

5 

37-i68 

62 

73 

68 

72 

65 

73 

61 

37.168 
















8 o6 


PHOTOGRAPHIC LONGITUDES. 


An inspection of the above set will show that a mean of a 
series is not likely to be in error more than .002 mm. or i" 
arc, as far as the actual measurements are concerned. 

As another example, a set of measurements for the radius 
of the moon may be given. For the quantity OP (diagram 
on p. 800) the following were the actual readings of the 
measurement: 


• 9 2 7 

23 

25 

24 

19 

14 

13 

13 

.924 

23 

27 

23 

19 

14 

13 

15 

•933 

30 

30 

21 

20 

*9 

12 

14 


It is at once obvious that the readings in the second half 
of each line are consistently smaller than those in the first 
half. The reason of this is found in the fact that the cross¬ 
wires of the micrometer were not truly at right angles, and 
consequently measurements taken in adjacent quadrants were 
not identical. The diaphragm carrying the cross-wires was 
rotated through 90° between each set of four measurements. 

To get a fair idea of the accuracy of these measurements 
we must, therefore, combine together the first and fourth in 
each line, second and fifth, etc. We then get the following 
values for OP: 


•923 

19 

19 

18 

.924 

l 9 

20 

19 

.926 

25 

2 1 

17 


Dividing by V2 — 1 to get radius, we have radius: 


2.228 

18 

18 

16 


2.231 

18 

2 1 

18 

2.223 

2.236 

33 

23 

H 

mean 


(0 An error in the local time will cause the same error in 
the resulting longitude, but an error in estimating the clock 
rate may have a somewhat more serious effect, inasmuch as it 
will alter the estimated interval between moon and star ex- 


SOURCES OF ERROR. 


So/ 

posures, and thus tend to produce an error in the moon’s 
right ascension and hence a large error in the longitude. 

U) Tllis source of error will be practically negligible, as it 
will only affect the longitude by the same amount. 

Suppose the observer systematically marked the exposures 
o. 1 second late. As this will apply equally to moon and star 
images, it makes the Greenwich time, and hence the longitude, 
o. 1 second wrong. 

(k) As the exposure on the moon is not instantaneous, 
the image will move to a slight, but quite appreciable, extent 
during the time the shutter is open. If the duration of the 
exposure be 0.2 second, the moon’s movement will be 3", and 
we should, therefore, tend to get a difference of this amount 
in the right ascension according as the image corresponds to 
the beginning or end of the exposures, i.e., according as the 
bright limb be following or leading. If the middle of the 
exposure corresponds to the recorded dock time, the moon’s 
right ascension will be in error by 1".$. Account must be 
taken, therefore, of the fact that the effective exposure is 
somewhat less than the total time the shutter is open, and it 
is probably not far wrong to reduce the amount by one-third; 
hence the error caused by the moon’s movement is not likely 
to exceed 1". This error changes its sign according as the 
bright limb be leading or following, and can, therefore, be 
completely eliminated by combining plates in pairs before 
and after full moon. 

(/) If a considerable time elapses between moon and star 
exposures, the change in refraction may become a serious 
source of error. To take the most unfavorable case: If the 
observer be near the equator and the photograph be taken at 
an altitude of 30°, a variation in temperature of io° Fahr. 
will cause a change of about *2". 5 in refraction, and therefore 
about that amount of error in the right ascension. Such 
unfavorable conditions as these would be very rare, and in 
general the possible error would be only a small fraction of 


8 o8 


PHOTOGRAPHIC LONGITUDES, 


this amount. This error disappears if the photograph be 
taken on or near the meridian, as in that case a change in 
apparent altitude will not affect the right ascension. It will 
also entirely disappear if the moon’s azimuth be used for 
computing the longitude, but this would only be of limited 
application, as, if the observer be on the equator, the azimuth 
is changing too slowly to ue ot any value. 

351. Precision of Resulting Longitude.—As a general 
conclusion it seems not unfair to state that there is no one 
source of error which should in any case exceed , unless it 
be the measurement of the moon’s radius. A limit of errors 
in this case is somewhat difficult to fix, but we shall probably 
be not far wrong if we assume a limit of double the above 
amount, and hence conclude that the right ascension of the 
moon can be determined within 4 seconds of time. As will 
be seen immediately, a higher degree of accuracy has been 
realized with actual plates. 

It is obvious that almost all the errors could be materially 
diminished if the method could be made a differential one, 
that is to say, if a duplicate photograph were taken with a sim¬ 
ilar instrument at about the same time at a fixed point. 

It now remains to give the results of plates exposed at a 
place of known longitude, which is as follows: 


Place , Chatham , England. True longitude , 2 m o8\I3 E. 


Plate. 

Date. 

No. Moon Images. 

Reference Stars. 

Long, from Plate. 

I 

1894, Oct. 16 

7 

a Tauri; 

Jupiter 

m. 

2 

s. 

07.1 

2 

i 1 a 11 

7 

( i 

t < 

2 

07.6 

3 

1895, May 2 

5 

(X Tauri; y Leonis 

2 

09.4 

4 

it c i it 

3 

t i 

c ( 

2 

06.0 

5 

1895, May 4 

5 

y Tauri; S Virginis 

* 

2 

07.0 


All the above plates were exposed with camera resting on 
a solid masonry foundation, and it is not probable that quite 
such accurate results would be obtained in the field. 













REFERENCE WORKS ON GEODESY. 


No attempt has been made in the following list of ref¬ 
erence works bearing on the subjects of geodesy and astronomy 
to include all those published which relate to the subject. 
The endeavor has been, however, to include those which have 
been consulted by the author in the preparation of this vol¬ 
ume, and a few others which have a particular bearing upon 
the subject. They are printed here that the reader may know 
where to look for more detailed information on the various 
branches touched upon in the text. 

Brainard, F. R. The Sextant. D. Van Nostrand & Co., New York. 1891. 
Chauvenet, William. Manual of Spherical and Practical Astronomy. 2 
vols. J. B. Lippincott & Co., Philadelphia, Pa. 1891. 

Clarke, Col. A. R. Geodesy. Clarendon Press, London. 1888. 

Cutts, Richard D. Field-work of Triangulation. U. S. Coast and Geo¬ 
detic Survey, Report for 1877. Washington, D. C. 

Davidson, George. Star Factors for Reducing Transit Observations. 
Report for 1894, U. S. Coast and Geodetic Survey. Washington, 
D. C. 

Douglas, E. M. Adjustment of Triangulation. School of Mines Quar¬ 
terly, No. 4, Vol. 14. Columbia College, New York. 

Dulles, Doctor Charles W. Accidents and Emergencies. P. Blakiston, 
Son & Co., Philadelphia, Pa. 1897. 

Eimbeck, Wm. Duplex Base Apparatus. Appendix No. n, Report for 
1897, U. S. Coast and Geodetic Survey. Washington, D. C. 
Freshfield, Douglas W, Flints to Travellers. Royal Geographic So¬ 
ciety, London, England. 1889. 

Gannett, Henry. Manual of Topographic Methods. Monograph 
XXII, U. S. Geological Survey. Washington, D. C. 

Gore, f. H. Elements of Geodesy. John Wiley & Sons, New York. 
1886. 

Hayford, John F. A Text-book of Geodetic Astronomy. John Wiley 
& Sons, New York. 1898. 

809 


8 io 


WORKS OF REFERENCE . 


Hilgard, J. E. Catalogue of Stars for Observations of Latitude. Ap¬ 
pendix 7, Report for 1876, U. S. Coast and Geodetic Survey. 
Washington, D. C. 

Hills, Capt. E. H. (R.E.). Determination of Terrestrial Longitudes by 
Photography. Monthly Notices, Royal Astronomic Society. Lon¬ 
don, January, 1893. 

Johnson, W. W. Theory of Errors and Method of Least Squares. John 
Wiley & Sons, New York. 1892. 

Lee, Thomas J. Tables and Formulae Useful in Surveying, Geodesy, and 
Practical Astronomy: being Professional Papers, No. 12, Corps of 
Engineers, U. S. A. Washington, Government Printing Office. 1873. 

Merriman, Mansfield. A Text-book on Method of Least Squares. John 
Wiley & Sons, New York. 1884. 

Merriman, Mansfield. Elements of Precise Surveying and Geodesy. 
John Wiley & Sons, New York. 1899. 

Oualbrough, Lt. E. F. Sailor’s Manual and Handy Book. 

Reber, Lt. Samuel. Manual of Photography. Government Printing 
Office. Washington, D. C. 1896. 

Runge, C. Photographic Longitudes. See translation, Appendix No. 
4, Report for 1893, U. S. Coast and Geodetic Survey. Washington, 
D. C. 

Safford, T. H. Catalogue of Mean Declination of 2018 Stars. Report 
Engineer Corps, U. S. A. Washington, D. C. 1879. 

Schott, Charles A., and Others. Geodetic Positions; Formulae and 
Tables for Computing. Appendix 9, Report for 1894, U. S. Coast 
and Geodetic Survey. Washington, D. C. 

Schott, Charles A. Determination of Time, Longitude, Latitude, and 
Azimuth. Appendix No. 14, Report for 1880, U. S. Coast and Ge¬ 
odetic Survey. Washington, D. C. 

U. S. Coast and Geodetic Survey. Geodetic Conference, Appendix 9, 
Report for 1893. Washington, D. C. 

Wharton, Captain W. J. L. Hints to Travellers. Royal Geographic So¬ 
ciety, London, England. 1889. 

Wisner, George Y. Geodetic Field-work. Trans. Am. Soc. C. E., Vol. 
XII. New York, July, 1883. 

Woodward, R. S. On Measurement of Base-lines. U. S. Coast and 
Geodetic Survey. Geodesy. Appendix 8, Report for 1892. Wash¬ 
ington, D. C. 1893. 

Woodward, R. S. Recent Experiences, etc., in Use of Long Steel 
Tapes for Measuring Base-lines. Trans. Am. Soc. C. E., Vol. XXX. 
New York, 1893. 

Wright, T. A. Adjustment of Observations. D. Van Nostrand & Co., 
New York, 1884. 


PART VII. 


CAMPING, EMERGENCY SURGERY, PHOTOGRAPHY. 


CHAPTER XXXVIII. 

CAMP EOUIPMENT AND PROPERTY. 


352. Attributes of a Skillful Topographer —The skill 
of a topographer is necessarily judged not only by his ability 
to perform his technical duties in the most efficient manner, 
but also by the time and cost of making the survey. The 
ultimate test of the ability of two men on the same class of 
work, providing each be equally skillful, is the relative cost 
of their work. If two topographers are able to make an 
equally good survey of the same territory, he is the more use¬ 
ful who performs the work the more rapidly and cheaply. In 
the U. S. Geological Survey extreme instances of this have been 
noted where two men have been engaged in the survey of the 
same region and have produced maps of equal quality on the 
same scale. One required, however, as much as three times 
as long to obtain this result as the other, and the work of one 
cost $7 per square mile, and that of the other $20 per 
square mile. This great difference was due in large measure 
to their relative skill in managing the parties, in the planning 
and control of the work of the assistants, and in subsisting 
and transporting the members of the field force. 


« 


811 



Si 2 CAMP EQUIPMENT AND PROPERTY. 

Where the topographer subsists on the country by living at 
hotels or farmhouses and hiring transportation, he need pos¬ 
sess but few accomplishments beyond those of a technical 
knowledge of his business and sufficient executive ability to 
enable him to properly direct the work of the members of his 
party. Where, however, he subsists in camp, additional 
knowledge must be his, of that common order Called “ horse- 
sense ” which comes to those who are brought up in the ways 
of the country and the woods. Finally, where the work 
takes him into inaccessible and unexplored regions he must 
possess, in addition to the qualifications already cited, a gen¬ 
eral knowledge of many things non-technical if he is to attain 
the relative success which would be looked for in similar work 
in other regions. 

Under these conditions he must have a general knowledge 
of all handicrafts with which he has to deal, for he will fre¬ 
quently find himself unable to rely upon the members of his 
party for the little matters of care and repair of outfit where 
the outside aid of skilled artisans is not procuiable. He will 
have to know not only how to harness a team and to adjust a 
saddle or pack (Fig. 201), but also how to repair a broken 
wagon or to shoe a mule, repair harness or make a riding- or 
pack-saddle. That he may make necessary repairs to the camp 
outfit and instruments, his equipment should include such tools 
as. will enable him to do various kinds of rough work in 
wood, leather, and metal. Many of these are enumerated in 
Article 367, but those which he may have to use in the repair 
of instruments he must select according to his own judgment 
and skill. 

An examination of the illustrations of the every-day life 
of the topographer in the far West (Figs. 186, 201 and 203) will 
give a clearer idea than words can convey of the nature of the 
travel and work which he will have to perform. Under such 
circumstances nearly all of the topographers of the United 
States Geological Survey have at one time or another had to 


5 UBS/S TENCE A ND ERA NSPOR TA TION. 


813 


repair or replace a broken tripod leg, a split plane-table board, 
an injured alidade or level. To replace cross-hairs or make a 
stadia rod are as common occurrences as the repair of a pair 
of old shoes, a torn coat, or a broken saddle-girth. 



Fig. 186.—Where a Pack-mule Can Go. 


353. Subsistence and Transportation of Party in Field. 

One of the most difficult problems connected with the execu¬ 
tion of topographic field-work is the subsistence and transpor¬ 
tation of the working force. Lack of judgment or experience 
in this figures largely in the output of the field force and the 
cost of obtaining a given result. Where the party can live 
on the country cheaply , that is, subsist in hotels or farmhouses 
and be transported by the people of the country, the work 
can be thus most economically managed. There are two 
modes of arranging payment for the services of the individuals 
of the party under such circumstances. One is to give them 
a per diem rate and to allow them to pay their own living 
expenses', the other, and that far more satisfactory where 
parties remain in the field a long time, is to pay the men by 


814 


CAMP EQUIPMENT AND PROPERTY. 


the week or month and to subsist and transport them. By 
this means the chief of the party has larger control of the 
movements and time of his working force. Subsistence may 
be had in various portions of the United States under such cir¬ 
cumstances at from $1.00 to $2.50 per day per man. Single 
conveyances may be hired at from $1.00 to $2.00 per day, 
including the feed of the animals, and from $2.00 to $4.00 
per day for a team with heavy wagon. 

The other mode of subsisting a party in the field is by 
camping, when tents, cooking outfit, animals and conveyances 
for transportation must be procured or the latter be hired for 
a period of time. This plan must necessarily be resorted to 
in many regions where habitations are widely scattered. To 
aid the party chief in selecting his outfit the following memo¬ 
randa have been prepared from a wide and varied experience, 
not only of the author in camping in various portions of the 
United States, Mexico, and India, but also from the expe¬ 
rience of his associates on engineering work on railroad and 
government surveys. 

354. Selecting and Preparing the Camp Ground. — 

When tents are to be pitched for a night or two only, it 
matters little where they are placed beyond choosing level 
and well-drained ground. When, however, the party is to re¬ 
main in the same camp for several days, much care should be 
exercised in selecting the best camp ground. There is much 
more in this than the mere choice of level holding-ground for 
the tents, and considerations which apply in one region are 
entirely reversed in others. 

In general the ground should be nearly level , having just 
slope enough to drain well. The soil should be preferably 
an open, earthy gravel, as this gives best underdrainage, 
holds tent-pins well, and is cleanly. Sand does not hold well, 
and loamy or clay soil is wet and damp after a rain, and when 
soaked will not hold the tent-pins. Moreover, it soon be- 


SELECTING AND PREPARING THE CAMP GROUND. 8l ‘ 

* 



Fig. 187.—A Pretty Camp Ground, North Carolina. 











TENTS. 


81/ 


comes filthy and tramped into mud-puddles about the camp 
animals which have to lie on it. The camp site should not 
be near a town because of the lack of privacy and the annoy¬ 
ance from visitors. Where cattle or hogs roam at large it 
should be in a fenced field. It should be especially selected 
for convenience to an abundant supply of good water (Art. 
378) and where fire-wood can be easily obtained (Art. 365). 

In the dense, damp woods of the North it should be in a 
clearing, and in the burning sunlight of the South and West 
it should be in the shade, preferably of a grove of trees (Fig. 
187). It should always be on slightly rising or high ground 
to assure good drainage, and as between camping in the bot¬ 
tom of a ravine or canyon or on top of a ridge, the latter 
should invariably be selected. There it is less damp and cold 
in early morning, and there the sunlight shines last at night. 
Care must always be exercised in selecting a camp site to 
choose one easily reached by wagons, pack-animals, etc., and 
which is convenient to forage or to pasture-land. 

355. Tents.- -The most satisfactory and comfortable tent 
for all general purposes is a p X 9 tent with <f-foot wa/l and 
extension fly (see left hand tent, Fig. 187). The best pattern 
of tents is that made by contract for the United States 
Army. Tents of fair quality, and nearly as good as those of 
the Army, are to be had of numerous makers in various parts 
of the United States. They should be of full 12-oz. army 
duck, warranted free from sizing and mildew-proof. All 
seams should be lapped at least one inch and double-sewed. 
The opening should be in the middle of one end and should 
be protected by flap of at least eight inches width which can 
be tied both inside and outside. Near the top of each 
end should be a small opening for ventilation and for insert¬ 
ing the pole (Fig. 188), and this should be protected in 
stormy weather by a canvas flap which can be tied down over 
it from the inside. Inside the tent, on a level with the top of 



818 


CAMP EQUIPMENT AND PROPERTY. 


the wall all around, at distances of two feet, should be tied 
strings, so that the wall can be raised in warm weather and 
tied up so as to allow the air to circulate freely. 

At the bottom of the wall should be a sod-flap of lighter 
duck and about eight inches in width. This is to keep out 
the wind and rain, so that when the tent is pegged down 
from the outside this flap is turned inside near the floor or 
ground, and kept in place by laying upon it dirt, short strips 
of wood or stone, or anything else which will weight it down. 
The use of dirt is not recommended as it is liable to rot and 



Fig. iS8.—Wall Tent with Fly. 


destroy the cloth. The ridge-pole should project about five 
or six feet in front of the tent, and be supported there by a 
third pole, and the fly should be this amount longer than 
the tent—that is, 14 to 15 feet in length, so as to extend 
like a porch as a shelter from sun and rain. An excellent 
modification of this extension fly is to so cut it that it will 
droop downward in a curved or turtleback form; then it will 
not be supported by an extension ridge, but will be merely 
guyed out by ropes. 

Frequently for convenience of transportation or for light¬ 
ness tents of other sizes must be used. In a large party it is 
often more convenient to have a 12 X 14 tent, in which a num- 
























TENTS. 


819 


ber of men may live or which may be used as a dining-tent. 
If for the latter purpose, such tent need not be provided with 
a fly. In very hot weather the most comfortable dining-tent 
is a simple fly which forms an awning as protection from sun 
and rain. Larger tents than 9X9 require at least two men 
to properly erect them. 

Where tents must be transported on the backs of men or 
animals, they must be of especially light design. It is then 
impracticable to carry poles unless these be jointed. Where 
wood can be procured, poles can be quickly hewn, or a rope 
may be used for the ridge and be tied to two trees, and the 
tent hung from these and guyed out as though supported by 
ridge and poles. Where timber is not convenient, light 
jointed poles may be carried, but the ridge can be dispensed 
with, a rope being used as a ridge by guying it out to 
some distance in front and rear. Under such circumstances 
small 7 X 7 A tents of 8-oz. duck and without flies may be 
carried. These can be had of weights as light as 6 to 8 lbs. 
to the tent, and will furnish shelter to a party of three or 
four men in the most inclement weather, providing they are 
properly ditched and have steep enough inclination to their 
sides to quickly shed water. 

In the tropics and where the heat is great, only the heav¬ 
iest tents will furnish comfortable protection. A light tent 
or one of moderate-weight canvas will not keep out the burn¬ 
ing rays of a tropic sun. At least two or even three tents 
of brown duck, each six inches to a foot smaller than the 
next, should be erected one inside the other so as to have an 
air-space of several inches about each; then if the wall at the 
bottom be raised to allow circulation of the air, the interior 
will be sufficiently cool for comfort. In India and Persia 
tents of light woolen cloth are used, and over the outside of 
these are placed tents of brown duck. These keep out the 
heat, retain the cool air of the early morning until late in the 
day, and protect from the rays of the sun. 


820 


CAMP EQUIPMENT AND PROPERTY. 


356. Specifications for Army Wall Tents —These and 
the following specifications are those issued by the Quarter¬ 
master’s Department of the U. S. Army to bidders on con¬ 
tracts. 

Dimensions. —Height, eight (8) feet six (6) inches; length of ridge, 
nine (9) feet; width, eight (8) feet eleven and one-half (11V2) inches; 
height of wall, three (3) feet nine (9) inches; wall eaves, two (2) inches 
wide; height of door, six (6) feet eight (8) inches; width of door, 
twelve (12) inches at bottom, four (4) inches at top; from top of ridge 
to wall, six (6) feet six (6) inches. 

Material. —To be made of cotton duck twenty-eight and one-half (28V2) 
inches wide, clear of all imperfections, and weighing twelve (12) ounces 
to the linear yard. 

Work. —To be made in a workmanlike manner, with not less than 
two and one-half (2 l / 2 ) stitches of equal length to the inch, made with 
double thread of five-fold cotton twine well waxed. The seams to be not 
less than one (1) inch in width, and no slack taken in them. 

Grommets. —Grommets made with malleable-iron rings, galvanized, 
must be worked in all the holes, and be well made with four-thread five¬ 
fold cotton twine well waxed. Sizes of grommets: For eaves, one-half 
(Vo) inch rings; for foot-stops, three-quarter (%) inch rings; and for 
ridge, three-quarter (%) inch rings; the latter to be worked so that the 
center will measure one and three-eighths (1%) inches from edge of 
roof, so as to be in correct position to receive spindle of upright poles. 

Door and Stay Pieces. —Door and stay pieces to be of the same ma¬ 
terial as the tent. Stay pieces on ends and ridge of tent to be six and a 
half (6y 2 ) inches square; those at corners of tent, at angle of roof and 
wall, to be eight (8) inches wide, let into the tabling at the eaves, and 
extending eight (8) inches up the roof and eight (8) inches down the 
wall; those on the sides to be four (4) inches wide and extending six (6) 
inches along the angles beneath the roof and six (6) inches along the 
walls. 

Sod Cloth. —The sod cloth to be of eight (8) ounce cotton duck, eight 
and three-quarters (8%) inches wide in the clear from the tabling, and 
to extend from door to door around both sides and ends of the tent. 

Tabling. —The tabling on the foot of the tent, when finished, to be 
two and one-half (2y>) inches in width. 

Ventilator. —An aperture four (4) inches wide and eight (8) inches 
long, one (1) in the front and one (1) in the back end of the tent, placed 
six (6) inches from the top and two (2) inches from the center, on the 
right side of each end. The aperture to be reinforced with eight (8) 
ounce cotton duck, and to have the edges turned in and stitched all 
around. A flap or curtain on the inside eight (8) inches wide and four¬ 
teen (14) inches long, finished, to be made of two (2) ply eight (8) 


SPECIFICATIONS FOR WALL TENT AND FLY. 821 


ounce cotton duck, stitched around the edges; to have one (i) “ No. i ” 
sheet-brass grommet placed at the top for the purpose of tying it up to 
close the opening; strings made of “No. 2” gilling line to be used for 
tying the curtain in place. 

Door Lines. —The door lines to be of six-thread manila line (large), 
three (3) feet long in the clear. 

Wall Lines— Eighteen (18) in number, to be two (2) feet long, to be 
made of “No. 3” gilling line, whipped at both ends and placed under 
the eaves on the seams, for tying the wall up. 

Door Fastening .—Door fastening, as shown in sample tent, to con¬ 
sist of four (4) double door strings of one-fourth ( 14 ) inch cotton rope 
one (1) foot long, on each side, passing through the door seam and 
secured by a “ Mathew Walker” knot. Brass grommets, “ No. 4,” to be 
in corresponding position on edge of door piece, in which to tie the 
door cords. A one and one-half (1V2) inch tabling to be made on the 
edge of door. 

Foot-stops. —Foot-stops, seventeen (17) in number, to be loops four 
(4) inches long in the clear, of nine-thread manila line, both ends pass¬ 
ing through a single grommet, worked in the tabling at seam, and to be 
held by what is known as the “ Mathew Walker ” knot. 

Eavc Lines. —Eave lines, ten (10) in number, to be of six-thread 
manila line (large), and to be eight (8) feet long in the clear, with an 
eye four (4) inches long, spliced on one end, and the other enn properly 
whipped and furnished with “ No. 3 ” metallic slip of Army standard. 

The tabling at bottom, the sod cloth, and the foot-stops to be so 
arranged that the sod cloth falls inside and the foot-stops outside the 
tent. 

All lines to be well whipped one (1) inch from the end with waxed 
twine, and properly knotted. 

357. Specifications for Army Wall-tent Flies. 

Dimensions. —Length, fifteen (15) feet and six (6) inches. Width, nine 
(9) feet when finished. 

Material. —To be made of cotton duck, twenty-eight and one-half 
(28%) inches wide, clear of all imperfections, and weighing ten (10) 
ounces to the linear yard. 

Tabling— A two (2) inch tabling to be worked on ends, and a one 
and one-half (1V2) inch tabling on sides. 

Grommets. —Grommets made with malleable-iron rings, galvanized; to 
be worked in all the holes with four (4) thread five (5) fold cotton twine, 
well waxed. Size of grommets for eave lines, one-half (V2) inch in 
diameter, and for upright spindle, three-fourths (%) of an inch in di¬ 
ameter; the latter to be placed so as to measure one and three-eighths 
(1%) inches from their centers to edge of fly, so as to be in proper posi¬ 
tion to receive spindle. 


822 


CAMP EQUIPMENT AND PROPERTY. 


Stay-pieces. —Stay-pieces on corners, triangular in shape, eleven (n) 
inches on base and perpendicular when finished, and on ridge six and 
one-half (6y 2 ) inches finished. 

Work .—The fly is to be made in a workmanlike manner in every 
respect, with not less than two and a half (2V2) stitches of equal length 
to the inch, made with double thread of five (5) fold cotton twine, well 
waxed. 

Seams .—The seams not less than one (1) inch in width and no slack 
taken in them. 

Rave Lines .—Eave lines, ten (10) in number, to be of six-thread 
manila line (large) and be seven (7) feet long in the clear, with an eye 
spliced on one end. four (4) inches long, the other end properly 
whipped, and furnished with a metallic slip No. 3, Army standard. 

All lines to be well whipped one (1) inch from the end with waxed 
cotton twine and properly knotted. 

358. Specifications for Army Wall-tent Poles. 

A set of poles to consist of two (2) uprights and one (1) ridge, th< 
former to be made of ash or white pine, and the latter of white pine 
clear, strait grained, and free from knots or other imperfections. 

Ridge .—Ridge nine (9) feet long, two and three-quarters (2%) inches 
wide, two (2) inches thick; on each end a band, two and three-quarters 
(2%) inches wide, of galvanized iron, secured by four (4) one and one- 


2" £& 

- . . 8 0 1 „II. . '••4 

0/ 



Fig. 189.—Ridge and Pole for Wall Tent. 


quarter (UA) inch copper nails. A five-eighths (%) of an inch hoie 
bored through at a distance of one and one-quarter (UA) inch from each 
end for the spindle of uprights. 

Uprights .—Uprights octagonal, ten (10) feet long and two (2) inches 
thick; band of galvanized iron, two and one-quarter (2%) inches wide, 
on upper ends, secured by two (2) one (1) inch screws. Spindle of one- 
half (Vs) inch round iron, galvanized, driven three (3) inches into upper 
ends and projecting four (4) inches. 

At set of pins for wall tent to consist of ten (10) pins twenty-four (24) 
inches double-notched, and eighteen (18) pins sixteen (16) inches single- 
notched. 


359. Specifications for Army Shelter Tents (Halves). 

Material .—To be made of Army standard cotton duck, thirty-three 
(33) inches wide, weighing from seven and one-half (7V2) to eight (8) 
























SPECIFICATIONS FOR ARMY SHELTER TENTS. §23 


ounces to the linear yard, and capable of sustaining a strain of seventy- 
two (72) pounds in the warp and thirty (30) pounds in the filling to the 
one-half (%) inch, counting not less than fifty-two (52) threads warp, 
and forty-eight (48) threads filling to the square inch. 

Dimensions and Workmanship .—To be about sixty-five (65) inches 
long on the ridge, and about sixty-one (61) inches wide when finished. 
The center seam to overlap one (1) inch. 

The four corners and center at the bottom of each half tent to be re¬ 
inforced with pieces of the same material firmly sewed on; said pieces to 
be about four (4) inches square when finished. 

The top or ridge to have a tabling, three and one-half (3V2) inches 
wide, and the bottom edge to be turned in and hemmed, making a three- 
eighths (%) inch seam neatly and securely sewed. 

To have two (2) grommet holes worked at each corner and at center 
of bottom. The two grommet holes at the top and front to be one and 
three-fourths (1%) inches to the center from the top, and the first hole 
one-half (V2) inch from the edge, and one and one-eighth (1%) inches 
from center to center apart. The two grommet holes at the top and rear 
to be one and three-fourths (1%) inches from the top; the first hole one- 
half (Vs) inch to center from the edge, and one and one-eighth (1%) 
inches from center to center apart. Along the bottom all the grommet 
holes to be one (1) inch from the bottom to the center, the first hole at 
the front one (1) inch from the edge, and about one and one-half (1V2) 
inches from center to center apart; the middle holes to be one and one- 
half (1V2) inches from center to center apart, and the rear holes to be 
one (1) inch from edge to center, and about one and one-half (iVs) 
inches from center to center apart. 

The closed end to measure three (3) feet eleven (11) inches from ridge 
to base, and three (3) feet seven (7) inches along the base; the grommet 
holes in this end to be one (1) inch from the edge and one and one- 
half (1V2) inches from center to center apart. All smaller-size grommet 
holes to be worked over a three-eighths (%) inch galvanized-iron ring, 



with two (2) ply of five (5) fold Cotton twine, well waxed; and the two 
(2) larger grommet holes made to receive the shelter-tent poles over a 
five-eighths (%) inch galvanized-iron ring, worked with two (2) ply of 
five (5) fold cotton twine, well waxed. To have nine (9) buttons and 















































824 


CAMP EQUIPMENT AND PROPERTY. 


buttonholes along the top, the first buttonhole to be one-half (V2) inch 
from the top, and three-fourths (%) inch from the edge; the others 
about seven and seven-eighths (7%) inches apart, with a white-metal 
button below each buttonhole, three (3) inches from the edge. 

The side of each half tent to have seven (7) buttons and buttonholes, 
the side buttonholes to be worked one-half (Vs) inch from the edge; the 
first buttonhole to be about eight (8) inches from the bottom, and the 
others spaced about seven and one-half (7V2) inches apart. The buttons 
to be firmly sewed on about three and one-half (3V2) inches from the 
edge. Along the closed end there shall be seven (7) buttons and button¬ 
holes, the first buttonhole to be about five (5) inches from the top, and 
all to be about one-half (y 2 ) inch from the edge, and spaced about six 
(6) inches apart; and the buttons to be about two and one-half (2%) 
inches from the edge, and spaced about six (6) inches apart. 

Each half tent to be furnished with a guy line, and four (4) foot-stops, 
made of 6-thread manila line, about one-fourth (Vi) inch in diameter; 
the former about six (6) feet seven (7) inches long in the clear with an 
eye-splice of about two (2) inches at one end; each foot-stop to be about 
sixteen (16) inches long in the clear, all whipped at both ends. All sew¬ 
ing, including buttonholes, to be done with W. B. linen thread, good 
quality, No. 70. 

360. Specifications for Army Shelter-tent Poles. 

A set of Shelter Tent Poles shall consist of two (2) uprights, made 
round, about one (1) inch in diameter, when joined to make a pole forty- 
six (46) inches in length from lower end to shoulder at top. with a neatly 
turned spindle at top about one (1) inch long and one-half (V2) inch in 
diameter, making a total length of forty-seven (47) inches. 




Fig. 191.—Jointed Shf.lter-tent Poles. 

1 

Each upright to be in two parts of about equal length, about two 
and one-half (2V2) inches bevel, and joined in a tin socket four (4) inches 
long, made of twenty-three (23) gauge tin (U. S. standard gauge), 
joined by a groove seam, neatly turned and soldered full length of seam, 
and secured to lower part of the pole by two (2) tacks, neatly and 
squarely driven. 

The pole to be of poplar wood, free from knots, and smoothly 
finished. 















ERECTING THE TENT. 


825 


361. Erecting the Tent. —To properly set up a tent it 
should be taken by the ridge and dragged away until laid out 
flat. The ridge-poles should be inserted through the ventila¬ 
tion-holes, the supporting poles inserted in the ridge-pole, 
and the whole raised and the corners at once guyed out. 
The corner ropes by which the tent is first stretched should 
be drawn in a diagonal direction so as to make an angle of 
about 45 0 with the walls. The door should be tied up so 
that the tent may be given its proper shape, and the wall- 
corner loops pegged down and door fastened to hold the 
whole in place. Then the side ropes should be guyed out 
and the tent stretched taut by tightening a little on each 
rope at a time. 

The fly must be laid over the tent when on the ground, 
and be raised with it. Then it must be so stretched as to 
touch the tent at no point excepting at the ridge, while at the 
eaves it should be from 6 to 10 inches above the roof of the 
tent. (Fig. 187.) This result can be obtained by several 
methods. One is to use pegs with two notches, on the lower 
of which the tent-guys are fastened, and on the upper the fly- 
guys; or an additional row of pegs may be set a foot beyond 
the tent-pegs for the support of the fly-guys. Where much 
rain or heat is encountered, short crotched poles about 10 
inches longer than the height of the wall should be cut and 
one of these be set under each of the corner fly-guys to raise 
the fly away from the tent roof. As a protection in high 
winds long guys should be stretched from each end of the 
ridge-pole in front and rear, otherwise storms blowing end on 
may carry the tent away. 

362. Tent Ditching and Flooring. —Where the ground is 
moist or rains are to be provided against, the tent must be 
ditched in order that: the water shall not run under it and wet 
the soil inside the tent; and where the camp is to remain in 
the same place for some time, the comfort of the party will be 
greatly increased by adding a floor to the tent. To ditch a 


826 


CAMP EQUIPMENT AND PROPERTY. 


DITCH 



tent a sharp spade or mattock should be used, and the soil be 
cut squarely or vertically just outside the foot 
of the wall. (Fig- 192.) The soil should be 
pitched away and an easy slope left on the out¬ 
side of the spade-cut. Dirt should never be 
banked up against the outer wall of the tent, as 
it rapidly rots and destroys the canvas. The 
ditch should be cut sufficiently deep to assure 
its carrying off any ordinary rainfall, and should 
Fig. 192.— Sod- b e m ade deep or shallow in various parts ac- 

CLOTH AND 1 1 

Ditch. cording to the slope of the ground, so that its 

bottom may have a uniform slope towards the lowest ground. 
At such point the ditch should be carried away from the tent 
a short distance in order to assure egress of the water from 
the ditch. 

The comfort of the occupants of the tent is increased by 
using a small strip of canvas or similar material as a floor on 
which to stand in dressing. Still better is a canvas floor of the 
full size of the interior of the tent, and this can rest upon the 
sod-cloth to keep out the wind. Where facilities for trans¬ 
portation permit, a wooden floor of tongue-and-grooved planks 
the length of the tent—say 9 feet—and fastened together by 
cleats in sections of 3 feet width may be provided. These 
3 X 9-foot sectional floors can be easily handled in moving, 
and the whole tent can be floored with them or only one or 
more sections be placed in the space between two cots. 

A still more substantial floor for a permanent winter camp 
consists in laying 2X4 scantling as floor-joists and planking 
these over so as to make a full floor which shall extend out¬ 
side the canvas walls. The rain will run under this, and a 
little carpet or canvas on it will keep the wind out. At each 
corner a 2 X 4 joist should be erected the height of the wall, 
and these corner posts should be connected by smaller scant¬ 
ling so as to form a railing the height of the wall. Over this 
the tent will be stretched, the framing of scantling holding 





CAMP STOVES, COTS , AND TABLES. 


827 


it out in shape. It is unnecessary except in very high 
winds to guy out tents stretched in this manner, the guys to 
the fly being sufficient protection. 

363. Camp Stoves, Cots, and Tables— The most com¬ 
fortable camp stove is the oil-heater. With this it is unnec¬ 
essary to cut any hole in the tent as an outlet for a smoke- 
pipe. It can be quickly lighted and extinguished, furnishes 
sufficient heat, and can be moved to any part of the tent with 
ease. Where oil cannot be carried for such a heater, a Sibley 
stove , made of sheet iron similar to that used in making stove¬ 
pipes, is one of the most simple and satisfactory heaters. 
This can be made by any tinner, is conical, the top being of 
the dimensions of ordinary small stove-pipe (Fig. 193), the 



A 



n n 

12% Circumference JL3^ Circumference 

193.— Tent Stove and Pipe. 

bottom 18 inches in diameter, and the height about 3 feet. A 
small hinged door must be cut and fitted in one side of this 
conical heater, the bottom being left open. In other words, 
it is an inverted funnel of stove-piping which rests on the 
bare earth. The sticks of wood are placed in it on end and 







































828 


CAMP EQUIPMENT AND PROPERTY. 


rapidly ignite and produce a strong heat. The fire is easily 
controlled by banking up the outer edge of the stove at the 
bottom, so as to prevent the ingress of the air, thus at once 
dampening it, or by digging away the earth underneath it a 
little, so as to admit the air, when the fire quickly draws up. 

The stove-pipe may be carried with a joint through one end 
of the tent. In this way the canvas will not be injured as 
when it is carried up straight through the roof, thus introduc¬ 
ing danger of fire from sparks and admitting rain around the 
pipe. In order to protect the canvas from burning by the 
heat of the pipe, a rectangular hole should be cut in the can¬ 
vas on three sides, the fourth side of the hole being left so 
that the canvas can be turned down, and when the pipe is 
removed it can be laid back again. On either side of the canvas 
surrounding this hole there should be fastened by rivets or 
wire thread a sheet of tin with a circular hole sufficiently 
large to permit the passage of the stove-pipe. 

The most convenient camp bed is a spring cot , where such 
can be transported, or one of the various forms of folding 
cots. Where for convenience of transportation cots cannot 
be carried and where hay can be procured, this or straw makes 
an excellent couch on which to lay the blankets, first placing 
canvas beneath these to protect them from particles of hay. 
In the pine and fir forests of the North a bed of boughs can be 
made by breaking off the small twigs from spruce or balsam 
boughs and laying these on end with the butt in the ground, 
their length not exceeding 12 to 16 inches. When enough of 
these are laid in this manner, one close against the other, they 
make a substantial, warm, springy mattress. A less comfortable 
couch of spruce or balsam boughs may be made by cutting these 
in lengths of 2 feet and laying them with the leaf ends to the 
center and the butts out, crosswise of the bed, in such man¬ 
ner that the butts project to either side. 

Various forms of folding camp tables may be purchased. 
These may, however, be made by a carpenter quite as conven- 


CAMP TABLES AND FIRES. 


829 


iently and cheaply and even more satisfactorily. Of those 
varieties which may be purchased, as convenient a form as any 
is the folding sewing-table one yard in length. For large 
tables the simplest is a pair of trestles on which to lay planks, 
and a similar arrangement may be provided as a bench on 
either side of the table. Of portable tables, one of the most 
satisfactory forms is that shown in Fig. 194, which may be 



readily constructed under the instructions of the topographer. 
This table is extensively used in the camps of the United 
States Geological Survey. The top consists of three 12-inch 
boards of suitable length screwed to two cleats. Each pair 
of legs is fastened at the table ends to wing boards 8 inches 
wide, and these are hinged to the top so as to fold inwards 
with the legs, thus lying flat to the table top. Lengthwise of 
the sides of the table top are hinged two other wing boards. 
When the legs are opened out these side boards are let down 
and hooked to the legs, thus keeping them in place. Long 
side hooks of iron add rigidity to the whole. 

364. Specifications for Sibley Tent Stoves. 

Stove. _The stove to be in the form of the frustrum of a cone, and to 

be made of No. 14 U. S. standard gauge common annealed plate-iron. 
To be in one piece (except the collar and door), and the seam at back to 
be fastened with twenty-four (24) rivets. The collar at top to be of the 
same material as the stove. To be two and a half (2 J / 2 ) inches deep, and 
be secured to the stove by six (6) rivets. Aperture for door to be about 
six (6) inches high by six (6) inches wide, the upper corners of which 
shall be rounded as in sample. The door to be sufficiently large to lap 
over the aperture; to be securely hinged to the stove, and to be properly 
molded to its form. An “ A”-shaped vent at the bottom of stove directly 
























830 CAMP EQUIPMENT AND PROPERTY* 

under the door, about two (2) inches high by three (3) inches wide; the 
top to be rounded. 

Dimension and Weight .—Height to top of collar, twenty-eight (28) 
inches. Circumference (outside) at bottom, fifty-eight (58) inches; at 
top, thirteen (13) inches. Distance from bottom of door aperture to 
base of stove, fourteen (14) inches. Weight about (19) pounds. 

365. How to Build Camp-fires. —To kindle a spark into 
a flame the spark should be received in a loose nest of the 
most inflammable substance at hand, which ought to be pre¬ 
pared before the tinder is lighted. When by careful blowing 
or fanning the flame is once started, it should be fed with 
little sticks or pieces of bark until it has gained strength to 
grapple with thicker ones. 

There is something of a knack in finding fire-wood. It 
should be looked for under bushes. The stump of a tree 
that is rooted nearly to the ground has often a magnificent 
root fit to blaze throughout the night. Damp or very sappy 
wood should be avoided. Dry manure of cattle is a fair fuel. 
Dry fuel gives out far more heat than damp fuel. Bones of 
animals also furnish a substitute for fire-wood. Wood should 
be cut into lengths of one foot and about two inches square. 
When nothing but brushwood is to be had, it should be 
burned in a trench. Where fuel is scarce, it is well when 
moving camp to gather and throw into the wagon all the dry 
wood which may be found along the road. 

366. Cooking-fire for a Small Camp —Lay down two 
green poles, 5 by 6 inches thick and 2 feet long, and spaced 
2 or 3 feet apart, and with notches in the upper side about 
10 to 12 inches apart. Lay two more green poles, 6 by 
8 inches thick and 4 feet long, in the notches. Procure a 
good supply of dry wood, bark, brush, or chips, and start the 
fire on the ground between the poles. The air will circulate 
under and through the fire, and the poles are the right distance 
apart to support a camp-kettle, frying-pan, or coffee-pot. 

If several meals are to be cooked in this place, it will pay 


CAMP EQUIPMENT. 


831 

to put up a crane. This is built as follows: Cut two green 
posts 2 inches thick and 3 feet long; drive these into the 
ground a foot from either end of the fire. If these poles are 
not forked, split the top end of each with the axe; then 
cut another green pole of same size and long enough to reach 
from one to the other of these posts; flatten the ends and 
insert them in the crotches or splits. The posts should be 
of such height that when this pole is passed through the bail 
of the camp-kettle or coffee-pot they will swing just clear of 
the fire. A less satisfactory crane is made by resting three 
poles together like a tripod and fastening them at the top by 
wire. Then a wire hook is hung from the center 'of these low 
enough to bring a kettle just over a fire built between the 
tripod legs. 

367. Camp Equipment. —For a party of six and where 
transportation is by wagon, the following covers most of the 
essentials of the living equipment for the camp—that is, the 
equipment exclusive of that required for transportation: 

Four 9 by 9 tents , with flies, poles, and pegs; one for party 
chief, one for three assistants, one for cook and kitchen, and 
one for dining and storage. 

Canvas or sectional wooden floors for tents. 

In winter, three heating-stoves, also one small (cast-iron) 
wood cooking-stove with pipes. 

Two mess-boxes , one for cooking-utensils, the other for 
tableware and light provisions, of pine screwed together, with 
hinged tops and compartments; also an inside cover the full 
width of the top, which may be used as a bread-board. 
When the lids are opened out and the two mess-chests placed 
together, they form a table of the width of the mess-chests, and 
a length four times their thickness. These chests should be 
20 inches deep, 20 inches wide, and 24 to 30 inches in length, 
so as just to fill a wagon bed. 

Mess-kit should consist of the following articles: 


832 


CAMP EQUIPMENT AND PROPERTY. 


2 wash-basins 
2 pepper-and-salt boxes 
2 buckets 
i dipper 

1 bread-pan 

2 frying-pans 
2 two-quart stew-pans 

1 half-gallon coffee-pot 
Table-cloths 
Dish-towels 
4 sheet-iron camp-kettles with cov¬ 
ers, sizes ranging from i to 3 
gallons so as to nest one within 
the other 

2 carving-knives 
1 spring-balance 

6 pans one and one half inches deep, 
six inches in diameter, for soup, 
oatmeal, etc. 


1 chopping-bowl and chopper 
1 iron broiler 
£ dozen cups and saucers 
kerosene-oil can 

1 dish-pan 

2 four-quart stew-pans 
10 plates 

1 quart tea-pot 
napkins 

2 one-quart cups 
1 coffee-mill 

$ dozen plated or aluminum table- 
knives, forks, table-spoons, and 
tea-spoons 

1 galvanized iron basting-fork and 
spoon 

3 pans two inches deep and eight 

inches in diameter, as serving- 
dishes 


All dishes, basins, etc., should be of granite- or porcelain- 
lined ware. The stew-, coffee-and tea-pots, etc., should also 
be of granite ware or have copper bottoms. To the above 
may be added numerous miscellaneous articles if transporta¬ 
tion facilities will permit, as a wash-tub and board, rolling- 
pin, etc. 

Where transportation is on the backs of animals tin and 



Fig. 195. —Folding Tin Reflecting Baker. 

galvanized iron will have to be substituted for granite-ware to 
reduce weight, and many of the above articles must be dis- 
















PRO VISIONS. 


333 


pensed with. The stove will be replaced for baking by a Dutch 
oven 12 inches in diameter, or by a tin reflector (Fig. 195). 

For transportation on men s backs practically everything 
will be dispensed with but a few tin plates and cups, knives, 
forks, and spoons, a coffee-pot, frying-pan, and stew-pan. A 
tin reflector should also be carried for baking. 

The miscellaneous camp tools may consist of some or all 
of the following: 


1 or 2 axes and extra axe-helves 
1 hatchet 
Bits and augers 
Screw-driver 

Assorted screws and nails 
Broom 

Quart canteens covered with cloth 
and canvas, or, in arid regions, 
a one-gallon canteen to each 
man 


Small files 

Rochester burners or other good 
lamps for drafting and reading 
Lanterns 

Assorted rope and string 

Whetstone 

Mattox 

Shovel 

Spade 

Saw 


368. Provisions. —The best estimate of the amount of 
provisions required for a camping party can be obtained by 
consulting the following ration list, which has proved most 
satisfactory after long experience in the field-work of the 
United States Geological Survey. 

A ration is the food estimated to be necessary to subsist 
one man one day. The amounts of the various articles in 
the ration are designed to be sufficiently liberal for all cir¬ 
cumstances. They are maximum amounts, which should not 
be exceeded. 

The Survey ration is made up of the articles and amounts 
given at the top of page 834. 

On the basis of this list a party of six will consume six 
rations a day. One hundred rations will therefore subsist 
such a party seventeen days. The cost of the above ration 
will vary necessarily with the locality. Near large markets 
and convenient to railways the ration—that is, the food 
of one man for one day on the above basis—costs from 45 to 


834 


CAMP EQUIPMENT AND PROPERTY. 


Table LXXI. 

RATION LIST. 


—- 

Article. 

Unit. 

IOO 

Rations. 

% 

Fresh meat, including fish and poultry (a) . 

Pounds 

IOO 

Cured meat, canned meat, or cheese (l>) . 

do. 

50 

Lard.. 

d ci, 

I c 

Flour, bread, or crackers. 

do. 

80 

Corn-meal, cereals, macaroni, sago, or corn-starch .. 

do. 

15 

Baking-powder or yeast-cakes. 

do. 

5 

Sugar. 

do. 

,lO 



Molasses. 

Oa 11 on «: 

I 

Coffee. . 

Pmi n ri c 

T 9 

Tea, chocolate, or cocoa. 

do. 

2 

Milk, condensed (c) . 

Cans 

10 

Butter. 

Poll nrl c 

TO 

Dried fruit (A) . 

do. 

20 

Rice or beans. 

d ci 

90 

Potatoes or other fresh vegetables (^). 

do. 

IOO 

Canned vegetables or fruit. 

Cans 

30 

Spices. 

O11 nres 

A 

Flavoring extracts.. 

do. 

4 

4 

Pepper or mustard. 

do. 

8 

Pickles. 

On a rts 


Vinegar. 

do. 

J 

T 

Salt.". 

Pounds 

A 

___ — -- 

4 


(a) Eggs may be substituted for fresh meat in the ratio of 8 eggs for i 
pound of meat. 

(i b ) Fresh meat and cured meat may be interchanged on the basis of 5 
pounds of fresh for 2 pounds of cured. 

(c) Fresh milk may be substituted for condensed milk in the ratio of 5 
quarts of fresh for 1 can of condensed. 

(I) Fresh fruit may be substituted for dried fruit in the ratio of 5 
pounds of fresh for 1 pound of dried. 

(c) Dried vegetables may be substituted for fresh vegetables in the 
j-atio of 3 pounds of fresh for 1 pound of dried. 


55 cents. It rarely exceeds 75 cents in the most inaccessible 
localities in the United States. 

Where transportation is difficult , as by pack-animals, the 
above must be varied by omitting the heavier provisions, 
those containing the most moisture, such as all canned goods, 
and these must be replaced by additional amounts of flour, 































RA TION LIST. 835 

beans, and dried fruits. Where fresh meat cannot be obtained 
it must be replaced by additional bacon and corned beef. 

Where provisions must be carried on men s backs a still 
further cut must be made in the heavier articles. Under the 
most unfavorable conditions an abundance of flour, bacon, 
rice, beans, oatmeal, cornmeal, tea, sugar, dried fruit, and salt 
must be provided. 

To the above ration list are to be added such quantities 
of matches and soap as may appear necessary. 


CHAPTER XXXIX. 


TRANSPORTATION EQUIPMENT. 

369. Camp Transportation ; Wagons. —The manner of 

transporting the camp outfit must depend necessarily on the 
conveniences of the country in which the work is being exe¬ 
cuted. On the plains or where there are sufficient roads, and 
forage can be provided for animals, transportation in heavy 
wagons is necessarily the most convenient and satisfactory. 
Even in the roughest country a large camp wagon with four 
animals will transport the outfit, including tents, beds, and 
provisions, for a party of six or eight. One of the most con¬ 
venient arrangements in hilly country is to have two smaller 
wagons, as they are more easily loaded and unloaded than a 
large one, and to have but one team to each wagon ; then on 
the heavy hills the teams may be doubled up until the sum¬ 
mits are reached. 

There should be bows to the wagon, that a canvas cover 
may be hung over these to protect the load from rain. 
Covers should not be laid on the load, as the latter will soon 
wear holes in it and render it useless. The load should be 
well tied down with a long quarter-inch lash-rope passed back 
and forth over the whole, otherwise the various articles will 
jostle about and wear holes or injure each other. Care should 
be taken in loading to place the heaviest and most durable 
property in the wagon bed, and the tents, bedding, etc., on 
top, especial care being taken that nothing which will wear 
holes in the tents shall touch them. 


836 


WAGONS—PACK ANIMALS AND SADDLES. 83 J 


There should be a tool-box in front of the dashboard to 
carry axle-grease, wrench, hatchet, wire, rope, nails, and sim¬ 
ilar articles which may be useful in event of a breakdown. 
The tailboard should be removed for easy loading, and in its 
place a long leather strap or chain be fastened about the 
mess-chests, which should occupy the rear of the wagon-bed. 
(Fig. 196.) 



370. Pack Animals and Saddles —The best pack-ani¬ 
mals are short-coupled, short-legged, stocky mules of less than 
one thousand pounds weight. A heavy load when moving 
camp at a slow gait is about 30 percent of the animal’s weight 
or between 250 and 300 pounds. Where the animal is to 
move at a trot it should be loaded with from 150 to 200 
pounds at the outside. A day’s journey for laden animals 
is 20 to 25 miles, and the best way of making the move is not 















































































838 


TRA NSP ORTA TION E Q U IP MEN T. 


to stop for a noon rest, but to set out early in the morning, 
continuing to the end of the journey without unpacking. 
Where longer trips have to be made, however, the packs and 
saddles should be removed and a full hour given for rest, 
otherwise the animals may be galled. 

The pack-saddle is fitted on the animal in the same man¬ 
ner as is a riding-saddle with a heavy saddle-blanket or pad 
underneath it. It is so constructed that it can be placed a 
little further forward than the riding-saddle. In addition to 
the ordinary saddle-cincha the saddle is sometimes provided 
with a crupper, but this is not as satisfactory for heavy work 
as is a breeching . The latter is made by screwing stay-straps 
to the rear end of each of the wooden pads of the saddle and 
carrying these back to a ring over the animal’s rump. Thence 
the stay-straps fall off to and support the forward ends of 
the breeching-straps as with ordinary harness. To the front 
ends of the breeching-straps are attached snap-hooks to catch 
into the cincha-rings on either side. 

In the Southwest the aparejo is generally employed as pack- 
saddle by the trained packer. This consists of two leather bags 
stuffed with hay which are connected at one end by a leather 
apron so that they may be hung over the sides of the mule. 
They are about 3 feet in length and 2 feet in width, and 
when fully stuffed about 4 to 6 inches in thickness. They are 
fastened to the animals by a wide girth thrown over the 
aparejo and under the animal’s stomach, and are firmly cinched 
on. Upon them is placed the load, divided into two parts of 
equal weight, one resting on each side of the aparejo and 
fastened in position by means of a long rope tied with a dia¬ 
mond hitch. This apparatus can, however, only be used by 
skillful packers, and is then rarely as satisfactory as is the 
Moore army saddle or the ordinary crosstree pack-saddle 
where the animals are compelled for any reason to move at a 
brisk gait. 

The crosstree pack-saddle can be purchased of dealers in 


AIOOKE PA CK-SA D OLE. 


839 


St. Louis, Denver, and similar supply centers. It can also be 
readily constructed. It consists of two pads of wood curved 
and shaped somewhat like the tree of an ordinary riding-saddle 
so as to fit the back of the animal, and these are joined 
together by two strips of oak or other stout wood screwed 
to the outsides of the wooden pads. They are fastened 
to each other at their junction, at which point they cross, 
one at front and the other at rear of the saddle. 

Panniers or alforjes of canvas, about 2 feet in length, 14 
inches deep, and 6 to 8 inches through, are hung on the saddle- 
forks by means of leather straps. To the outer sides of these 
are fastened at one end a long leather thong, and at the other 
a loop or metal ring. The thongs are thrown across the back 
of the animal, passed through the loop on the opposite alforje, 
and tied up so as to raise the load on to the tree of the saddle 
and away from the sides of the animal. The center of the 
load may be filled with loose and light articles, as blankets 
or a tent, to give the whole shape and body. Over this 
should be thrown a canvas cover, and the pack be tied on by 
means of the diamond hitch. The lash-rope which fastens 
the load on the aparejo or pack-saddle should be of -J inch 
manila rope 42 feet in length. This should be fastened to a 
wide girth of canvas which comes under the belly of the ani¬ 
mal, and on the other end of the girth should be an iron hook 
or a hook made from the crotch or forked branch of some hard 
wood. 

371. Moore Pack-saddle. —The United States Army 
uses an improved saddle (Fig. 197) for packing which is a 
modification of the Mexican aparejo, over which it has several 
advantages, chiefly in that it is more easily handled by in¬ 
experienced packers and is more readily kept in good condi¬ 
tion. The Moore saddle, as it is called after its inventor, 
consists, like the crosstree saddle, of a number of parts, 
including the saddle proper, two pads similar to those of the 
aparejo, a crupper instead of the breeching used with the 


840 


TEA NSP OE 7'A T10N E Q U IP MEN 7\ 


crosstree saddle, a corona or pad placed next to the animal’s 
back under the pad, and a large canvas pack cover; also 



Fig. 197.—Full-rigged Moore Army Pack-saddle. 

a canvas cincha ten inches in width, of varying length ac¬ 
cording to the animal (Fig. 198); half-inch manila rope 



Fig. 198.—Pack-saddle Cinches. 


twenty-two feet long for sling-rope, and a lash-rope similar 
to that used with the crosstree saddle. 

The army saddle is adjusted to the animal somewhat dif- 


























































THROWING THE DIAMOND HITCH . 841 

ferently from the crosstree saddle. The cincha goes entirely 
over the saddle, coming under the animal’s belly and over his 
back, thus completely encircling or girdling him and the 
saddle. The pack is loaded in a manner similar to that 
described for the crosstree saddle. 

37 2. Throwing the Diamond Hitch.—It requires two 
men to lash a pack with the diamond hitch unless the packer 
possess unusual skill. Calling the two packers, respectively, 
thrower and cincher, the latter stands on the off or right side 
of the animal, and the former on the left or near side. 
The thrower first casts the girth under the animal’s body to 
the cincher, who grasps the hook, point to. front, in his left 
hand. The thrower immediately casts the end of the rope 
backward over the left shoulder of the animal across his rig-ht 

o 

hip, then taking the short or girth end in his right hand and 
the long or loose end in his left hand—that is, the end toward 
the head of the animal—he casts a short loop of the rope over 
the back of the animal (Fig. 199, A) to the cincher, who 
passes this through the hook of the girth and draws it slightly 
taut. The cincher at the same time throws the remainder of 
the slack of the rope over the back and towards the head of 
the animal and on the side of the thrower (Fig. 199, B). 

The thrower next passes the long end of the loop backward 
over the rope first cast, thus making a bight in it, and he also 
carries it in front of the forward corner of the pack on his 
side, leaving the short end hanging backward—that is, to his 
right (Fig. 199, C). 

The cincher now takes his slack loop in his right hand, 
and reaching beyond the bight just made by the thrower in 
the cinch rope, he passes his loop backward under and forward 
over the cinch or first rope, thus making a second bight in 
it. Immediately he takes the loose end, which is that to his 
left, and passes it forward and across the pack under the rope 
which he has just looped (Fig. 199, D ). 

The cincher now takes the end of the rope which is caught 


842 


TRA NSPOR TA T10N E Q UIP MEN T . 


in the hook and, pressing his foot against the side of the ani¬ 
mal, draws this as taut as he can, while the thrower, turning 
his back to the pack, takes in the slack by holding it tautly 
over his shoulder. This slack he passes over the front corner 

Tail. 



Head. 


Fig. 199.—Lashing Pack with Diamond Hitch. 

of the pack and, still holding it firmly, passes it under the 
same and backward, pressing his foot against the rear corner 
of the pack to draw it as taut as possible (Fig. 199, E). Then 
the cincher, standing to the rear of his side of the pack, takes 
in the slack given him by the thrower and, pressing his foot 
against the rear of his pack, draws the rope as taut as possible. 























































L0AD1XG PACK-MULE WITH MESS-BOXES . 


^45 



Fig 2qc —Loading Pack-mtle with Mess-boxes. 




















r 






















* 























PA CKING 


ON MAN’S BACKS , AD1R0NDA CKS. 


845 



Fig. 2oi.—Packing on Men’s Backs, Adirondacks, 








PA CKMEN. 


847 


The slack he passes around the rear end of the pack on his 
side, under it and up the forward side, pressing his foot 
against the pack from the front, while the thrower, using his 
foot against the front side of the pack on his side, takes in the 
slack given him by the cincher (Fig. 200). 

Having the entire pack now fastened, it will be noted that 
the two bights open the loops in the form of the diamond 
hitch. The thrower then takes the slack end, which he now 
holds, and ties it on his side across the front and outer side 
of the pack in such manner as to firmly bind the whole to¬ 
gether (Fig. 199, F). 

373. Packmen.—Where camp equipment must be trans¬ 
ported on men’s backs, as in some portions of the Adiron- 
dacks (Fig. 201), in the Northwest, and in Alaska, the loads 
may be arranged thus: Blankets, clothing, etc., may be rolled 
inside of rubber or canvas into bundles of about 24 inches in 
length, 18 to 20 inches width, and 15 inches thickness. These 
should be strapped and slung over the shoulders by wide 
leather straps fitted in a manner described below for pack- 
baskets. For heavy provisions and miscellaneous small ar¬ 
ticles baskets of the type used in the 
Adirondacks, or canvas panniers, furnish 
the most satisfactory mode of carrying 
packs. 

These baskets are shaped as shown in 
Fig. 202, averaging about 18 inches in 
depth, 17 inches in width at the bottom, 
and 15 inches in width at the top, with 
the thickness at bottom and top 12 
inches. A heavy leather strap is run 

Fig. 202.—Pack-basket. 

around the top under the rim, and to 

this are attached two carrying-straps which come close together 
and pass through the same loop at the top. These straps pass 
down the body side of the basket close to the latter, and are 
caught up at the bottom of the basket at their outer extrem- 









848 


TRANSPORTA TION EQUIPMENT. 


ity, so as to form the letter “A” as viewed against the 
basket. Thence they run up and buckle to the ends which 
come from the upper portion of the basket, leaving wide 
loops through which the arms can be passed, while the 
buckles give necessary freedom for adjustment. This pack 
should be carried as high as possible on the shoulders, and 
the closeness of the straps at the top keeps it well on to the 
shoulders without a confining breast-strap. 

A heavy load for a packman over good trails and for 
tramps of 15 to 20 miles is 60 to 75 pounds. A light load 
for heavy traveling and mountain work is 35 to 50 pounds. 

374. Transportation Repairs.—In addition to camp 
wagon and harness or pack-saddles, as the case may be, the 
following should be carried for repair and use in connection 
with the animals and outfit. 

A farrier s kit for shoeing where blacksmiths cannot be 
had: this should include one clinch-cutter, one clinching- 
iron, one shoeing-hammer, one pair shoeing-pincers, one 
shoeing-rasp, assorted horse or mule shoes already fitted and 
corked, and assorted nails. 

A saddler s kit , tor use with pack and saddle outfits: this 
should include sewing-palm, bradawls, sail-needles, twine, 
wax and sewing-thread, assorted buckles, assorted copper 
rivets, rivet and iron set, riveting-pincers, rivet-nippers and 
cold-chisels, assorted rings, leather whangs, lace leather, also 
some heavy harness leather and copper wire. 

In addition to the above the following miscellaneous utensils 
should be provided : 

Axle-grease, nose- or feed-bags, horse-brush, currycomb, 
halters, whips, riding saddles and bridles, saddle-blankets, 
wagon-jack, monkey-wrenches, and canvas pack-covers. 

To the above may be added, under certain conditions: 
hopples, bells, tethering-ropes and long pivot picket-pins 
with rings at top, water kegs or barrels, heavy canvas wagon- 


VETERINARY SURGERY. . 849 


covers with bows for supporting same, lash-ropes for tying 
packs on wagons. 

The amount of forage required by animals doing heavy 
work may be estimated roughly from the following: 


14 lbs. hay or fodder, 
12 qts. oats, or 
8 qts. corn 


r 

per horse 

v "< 

per day 


Hay, when pressed, 11 lbs. 
to cubic foot, 32 lbs. 
to bushel, 25.71 to 
cubic foot. Grain 56 
lbs. to bushel, 45.02 
to cubic foot. 


375. Veterinary Surgery —Some general remedies should 
be carried for the use of the animals. These may consist 
chiefly of the following: 

Liniments of ammonia or strychnine for external applica¬ 
tion, as for sprains, by reducing heat without blistering. Soap 
liniments and iodine compounds for external application to 
swellings. 

Cleansing agents for decomposing sores, consisting of sul¬ 
phate of copper or bluestone or of carbolic acid in weak 
solutions. All excellent for curing “scratches.” 

Astringents to diminish the discharge of wounds, as alum 
or sulphate of zinc. 

Healing agents for wounds, as collodion and arnica. 

Emollients to soften and relax muscles, as olive oil and 
poultices. 

Cathartics, as Epsom salts, castor-oil, aloes. 

Stimulants for the stomach, as ginger, gentian, and caraway 


seeds. 

For cramps salicylic acid, oil of turpentine. 

Diuretics for bladder and kidneys, as turpentine, sweet 
spirits of niter. 






CHAPTER XL. 


CARE OF HEALTH. 

376. Blankets and Clothing —The personal property to 
be carried by each individual of the party will depend neces¬ 
sarily, as do the other articles of the camp outfit, upon the 
mode of transportation and the region in which the party is 
to work. Where wagon transportation is provided and the 
party may carry all essentials, each individual may take a 
small steamer-trunk for his clothing and should roll and strap 
his blankets in the form of a cylindrical bundle in a piece of 
No. 6 canvas or 16-oz. duck. The canvas cover should be 7 
feet long b*y 6 feet wide, so that when the bed is laid down it 
may rest on half of the canvas to keep out moisture if on the 
ground, and air if on a cot, and the other half of the width of 
canvas should be passed over the bed to protect it from air 
and moisture. 

An excellent mattress consists of a good large comforter 
folded three times endways, the width being about as long as 
a man’s body. Additional wool blankets and a comforter 
should be taken for covering, the number depending upon the 
climate! Sleeping-bags , such as are now sold by dealers in 
sporting goods, furnish the warmest and most comfortable 
bed for almost any condition of camping. 

Where rain is to be encountered a mackintosh or rubber 
coat is of little value. A heavy oil-slicker is the most 
serviceable garment; and for horseback, leggings and pea- 
jacket of oil-slicker. The most serviceable hat is a heavy, 
wide-brimmed soft felt sombrero or army campaign hat for all 
climates and conditions. The more intense the heat of the 

850 


BLANKETS AND CLOTHING. 


■851 


sun’s rays and the more penetrating, as in the tropics, the 
heavier should be the head-covering. Under such circum¬ 
stances a heavy pith helmet may be used; but, be helmet or 
heavy felt sombrero worn, a band made of light linen or India 
silk, folded to about three inches in width and of a length of 
about two yards, should be wrapped around the hat close to 
the brim so as to make a thick pad over the temples to keep 
out the penetrating rays of a tropic sun. 

Rubber boots should never be used even in snow or water. 
In deep snow or intense cold, arctics or a wrapping of gunny- 
sack over the leather shoe may be employed. For climbing 
a shoe is much more comfort¬ 
able and supple than a boot. 

For riding leather leggings 
may be added, or else water¬ 
proof leather boots may be 
used. In any event, in cold 
or wet and for heavy climbing 
water-proof leather shoes with 
thick extension soles should 
be worn (Fig. 203). In the 
tropics the foot-covering 
should be light and supple, 
but the soles should be heavy 
both to protect the feet from 
moisture and to keep out the 
heat of the soil. Light canvas 
shoes should be carried to rest 
the feet, which easily blister in tropic lands. 

Where much foot-work is done, very heavy, coarse cotton 
socks should be worn. In cold weather heavy woolen socks, 
and in intense cold and deep snow German felt socks, must be 
worn. As a substitute for leather boots and arctics, felt boots 
may be worn over the German socks. 

In high altitudes the underwear should always be of heavy 



Fig. 203.—Plane-table Station 
on Mountain in Alaska. 








852 


CARE OF HEALTH. 


wool regardless of how high the temperature may be in the 
daytime. Medium-weight wool may be carried, and two suits 
be worn, one over the other, in very cold weather. The sudden 
changes at evening and night render heavy underwear an 
essential to health. In the tropics light silk gauze or a mix¬ 
ture of silk and wool underwear should be worn next the skin 
to absorb the moisture of the body. 

Fo x sleeping-clothes, pajamas only should be used, and in 
the tropics especially these should be of light flannel. Also 
in the tropics flannel cholera-bands should be invariably worn 
over the abdomen, and never removed except to change. 

For work in the brush or woods the most satisfactory 
outer garments are made of brown duck, or light overalls may 
be pulled on over woolen trousers. In cold and windy 
weather, such as is experienced at high altitudes, flannel-lined 
hunting-coats and trousers of duck should be worn, the duck 
keeping out the wind. The best coat for wind protection is 
the blanket-lined leather hunting-coat. A canvas or leather 
hunting-coat, lined or unlined, is a most convenient garment 
for the surveyor because of its numerous pockets. In the 
cold and at high altitudes a woolen sweater should be carried 
or worn in preference to an overcoat (Fig. 203). 

In addition to the above the novice in camping should not 
neglect to take towels, soap, and miscellaneous toilet articles. 
Where the party is to sleep in the open air, or when the weather 
is very cold, the head should be covered with a knit nightcap 
of worsted, the most satisfactory being the conical toboggan 
cap, which can be pulled down well over the ears and head. 

The lack of a tooth-brush, even in the Arctics, has been 
known to produce sore mouth and gums in one accustomed 
to its use. Toilet-paper is essential, especially in extremely 
hot or cold climates, or piles may result. Camping is at best 
uncleanly, and every effort should be made to keep the person 
and the camp as clean as possible. Even then much dirt will 
have of necessity to be encountered. 


GENERAL HINTS. 


*53 


377 - General Hints on Care of Health. —In camping or 
working in high altitudes the topographer is liable to contract 
a disease known as mountain fever , which is allied to typhoid. 
It is caused chiefly by carelessness in becoming overheated 
under the hot rays of the midday sun and then suddenly 
chilling off in the night air. The precaution already de¬ 
scribed of covering the head at night and of wearing heavy 
woolen underwear, despite the intensity of the heat at midday, 
will generally suffice to protect the camper from any sickness 
in the healthful climate usually found at high altitudes. 

In the tropics the traveler is liable to sickness from mala¬ 
rial fevers, dysentery, and cholera. With proper attention to 
food and clothing, if living a healthful outdoor life, one is 
hardly more liable in the tropics than elsewhere to contract 
other diseases than malaria if great care is exercised in carry¬ 
ing out the following suggestions: 

Wear thick-soled shoes of soft leather, and change or dry 
these, going barefooted meanwhile if necessary, as soon after 
they become wet as practicable. In other words, do not keep 
wet shoes on the feet, and do not wear rubber to protect 
against moisture. If the body become wet from rain or ford¬ 
ing streams, the clothes should be taken off and dried as soon 
as possible thereafter. The body should never be allowed to 
steam while covered with drying shoes or clothes. 

Flannel cholera-bands should be worn at all times. Cloth¬ 
ing worn in the daytime should invariably be changed at 
night for flannel pajamas. The head-covering must be of the 
heaviest, and the protection over the temples should be espe¬ 
cially heavy. The topographer should not expose himself 
to the direct rays of the sun more than absolutely necessary, 
and where practicable should be shaded by an umbrella. The 
back of the neck should be shaded from the level rays of the 
early morning and late afternoon sun by a cloth veil hung 
from the back of the hat. 

The camper should sleep in a hammock or on a cot. He 


854 


CARE OF HEALTH. 


should, if possible, never go to sleep wet or on wet ground, 
and when this is unavoidable he should endeavor to sleep in 
dry woolen blankets, or, if he must sleep in wet blankets, 
these should be of light wool and should be next his body. 
Above all, the head should always be protected from the 
night dews either by some temporarily improvised shelter, by 
covering with a sheet, or the canvas bed-cover, or mosquito¬ 
netting fine enough to keep out the moisture. He should 
avoid rising before the sun has dispelled the night dew. 
Early rising is very dangerous in malarious regions. 

Where possible, drinking-water should always be boiled 
and allowed to cool. (Art. 378.) At work it is best to carry 
in the canteen boiled water or thin coffee or tea. Lime-juice 
should be freely used in water which is not boiled. Weak 
ginger tea made of a thin effusion of Jamaica ginger with a 
little sugar is a palatable and safe beverage, especially where 
the water is alkaline. Unless absolutely unavoidable, water 
which is standing in the sun, especially running water in shallow 
streams, should never be drunk without previously boiling or 
adding whiskey or lime-juice to it. Water should be kept 
shaded from the sun as far as practicable, and only water 
which has stood overnight to cool should be used if possible. 

Water may be kept fairly cool in canteens throughout the 
day if they are heavily covered with one-half inch of woolen 
blanketing shielded outside by heavy canvas. This covering 
should be soaked in the morning, and as it evaporates it keeps 
the canteen water cool. When it dries off it should if possi¬ 
ble be again soaked, perhaps several times during the course 
of the day. The covering should be omitted on the edges 
under the carrying-strap. 

Heavy foods and flesh foods should be used sparingly. 
Fresh meats once a day and in moderate quantities should be 
eaten to keep up the system, but not more than one such meal 
a day should be consumed. Jerked or sun-dried meat, chipped 
beef, or the “ carne seca ” of Spanish America will not pu- 


DRINKING- wa ter. 


855 


trify under the most unfavorable circumstances, and make a 
palatable dish when stewed with canned corned beef, potatoes, 
onions, or other vegetables. Bacon may, in spite of the fact 
that it contains fat, be used once a day. Eggs should not 
be indulged in too freely. Cereal foods, as rice, cornmeal, and 
good bread, beans and peas, should be used freely. 

Fresh fruit should be used most carefully and sparingly. 
It may be safely eaten in the morning providing it has been 
picked overnight and allowed to cool in the night air. It 
should never be eaten after ten o’clock in the morning, not 
only because of the heat of the body, but also because the 
fruit itself is hot. It is most dangerous when picked ripe 
from the tree in the hot sun. Not only over-indulgence but 
any indulgence in fresh fruit after the heat of the day has 
come on is most dangerous. Fruit which has been kept on 
ice or otherwise cooled may be eaten sparingly after sun¬ 
down. 

Excess in drinking or eating should be scrupulously avoided 
in all climates. Alcoholic liquors should never be indulged 
in, especially in the tropics, excepting for medicinal purposes. 
Prolonged immersion in bathing should be avoided in all cli¬ 
mates, especially if the water be cold. A quick plunge or 
sponge-bath may be indulged in daily in early morning or late 
evening. 

378. Drinking-water. —Nothing is more certain to secure 
endurance and capability of long-continued effort than the 
avoidance of everything as a drink except cold water, and at 
breakfast a little coffee. The less drunk of these on a long 
tramp the better, since one suffers less in the end by control¬ 
ling the thirst, however urgent. 

Poisonous matter of many descriptions may be taken into 
the stomach in drinking bad water. Dysentery and malarial 
diseases ensue from its use. With muddy water the remedy 
is to filter; with putrid water , to boil, to mix with charcoal, or 
expose to the sun and air, or, what is best, to use all three 


856 CARE OF HEALTH\ 

methods at the same time. With salt water nothing avails 
but distillation. 

Sand, charcoal, sponge, and wool are the substances most 
commonly used in filtering muddy water. A small piece of 
alum or, better, powdered alum is very efficacious in purifying 
water from organic matter, which is precipitated by the alum, 
and left as a deposit in the bottom of the vessel. Above all, 
whenever there is the least uncertainty as to the quality of 
the water, boil it. Nothing is so sure a preventive of sick¬ 
ness in camping in warm climates as the exclusive use of 
boiled water for drinking. 

379. Medical Hints.—As the topographer and the ex¬ 
plorer are frequently so circumstanced as to be unable to 
promptly procure proper medical attention, and as the nature 
of their duties is such as to render them liable to certain 
classes of sickness and to violent injury, the following sugges¬ 
tions have been prepared for the emergency treatment of the 
sick and injured. In all cases of serious illness or of fractured 
limbs the best medical advice procurable must be sought at 
once, however far it may be necessary to seek it or to move 
the patient. In Article 384 is given a list of the most useful 
emergency medicines with their uses and size of dose. 

Malarial Fevers .—These are, of all diseases, the most 
likely to be contracted in camping in semic-tropic and tropic 
regions. They should be treated by administering 15 to 20 
grains of quinine before the expected attack. This should 
be preceded invariably at first by one to two compound 
cathartic pills. If the dose be given twelve hours previous to 
the renewal of attack, it will have better results. In malarial 
localities a tablespoonful of whiskey with 4 to 6 grains of 
quinine should be taken daily as a tonic. In severe cases of 
malaria there should be given, excepting in the hot stage, 
quinine in doses of 15 grains at intervals of eight hours. A 
Dover’s tablet should be given every three hours with quinine 
in obstinate cases of malarial fevers. Wherever possible, 


DIARRHEA AND DYSENTERY. 


857 


even at the expense of suffering to the patient, he should be 
removed to a higher and dryer situation, if such be accessible. 

Colic is treated by giving ounces of castor-oil with 20 
drops of tincture of opium. Also it may be treated with 10 
drops of essence of peppermint or a teaspoonful of Jamaica 
ginger in hot water. Hot turpentine fomentations should be 
applied to the abdomen, and 3 grains of calomel and soda 
may be given instead of the castor-oil. 

For Constipation give compound cathartic pills, a saline 
purgative, as Epsom salts, or two tablespoonfuls of castor-oil. 
In obstinate cases, enemas of warm water with olive or castor 
oil or castile soap should be given, the patient meantime 
lying down. 

For Frostbite moderate friction should be opplied to the 
parts affected. They should not be warmed until recovery 
is well advanced. Where snow is procurable, friction should 
be produced at first with this or with sponges dipped in ice- 
water. As the parts become warmer and less congested they 
should be encased in dry flannel or cotton wool. 

In all cases of Poisoning vomiting should be at once en¬ 
couraged. The simplest ways in which to induce it are by 
large draughts of lukewarm mustard-water, ipecacuanha, soapy 
water, or by tickling the throat from the inside. After this 
soothing liquids should be administered, as beaten raw egg, 
flour and water, or milk in large quantities. If the sufferer 
be much depressed and have cold hands or feet, and blue 
lips, some stimulant may be administered, preferably strong 
hot tea or coffee. 

380. Diarrhea and Dysentery.— Errors in diet resulting 
in simple Diarrhea may be treated with a mild laxative of 
castor-oil or cathartic pills. A change of diet should be 
made to milk and well-boiled arrowroot. A glass of port 
wine and brandy with plenty of sugar and nutmeg may also 
be administered occasionally, and the patient be kept as 
quiet as possible. If diarrhea refuses to yield to the above, 


858 


CANE OF HEALTH. 


take 3 grains of calomel and soda at a dose. Should it occur 
after a chill or in localities where dysentery is prevalent, 20 to 
30 drops of chlorodyne should be given, followed at bedtime 
by five Dover’s powders. 

Though one of the most feared of all tropical diseases, 
Dysentery yields quite readily to timely treatment. It is most 
commonly caused by sudden or prolonged chills, or results from 
bad drinking-water or food. Symptoms are diarrhea followed 
by irregular and shooting, griping pains, straining and discharge 
of mucus from the bowels. As the disease advances the pains 
are more distressing and the actions more frequent, the dis¬ 
charge being tinged with blood and of most offensive odor. 

This is the first stage of the disease, the treatment of 
which consists of immediate rest in bed and turpentine fomen¬ 
tations on the abdomen followed by a large linseed poultice. 
A mustard-leaf should be placed on the pit of the stomach 
and 20 drops of tincture of opium in water be administered, 
followed by 20 to 30 grains of ipecacuanha powder mixed in 
water. Fluids should be abstained from to avoid vomiting. 
Repeat ipecacuanha powders twice at intervals of six hours, 
and give five to ten grains of Dover’s powder at bedtime. The 
food during and for some time after the disease should con¬ 
sist of boiled milk, weak meat broths, and well-boiled arrow- 
root. Beef tea should be avoided as too heavy. 

Where malaria is present 15 grains of quinine may be 
given in addition to the above. In advanced cases Dover’s 
powders should be given instead of ipecacuanha. Diet is all- 
important in this dread disease, as the smallest particle of 
solid food may set up an irritation which will prove fatal. 

381. Drowning and Suffocation. —Drowning may some¬ 
times be of such duration as to cause natural breathing to 
cease. Treatment consists in the re-establishment of the action 
of breathing by means of artificial respiration. The body 
must be at once freed from clothing which binds about the 
neck, chest, and waist and b z turned on the face , a finger being 


DROWNING AND SURGERY. 


859 


thrust into the mouth and swept around to remove anything 
which may have accumulated there. Respiration may then 
be restored by Sylvester’s method, which is as follows: 

The body is laid out flat on the back , with a folded blanket, 
shawl, coat, or stick of wood under the shoulders, so as to 
cause the neck to be stretched out and the chin to be carried 
far away from the chest. The tongue is drawn carefully 
forward out of the mouth by holding it with a cloth. 

Some one now places himself on his knees behind the head, 
seizes both arms near the elbows, and sweeps them round 
horizontally, away from the body and over the head until 
they meet above it; when a good, strong pull is made upon 
them and kept up for a few seconds. This effects an inspira¬ 
tion —fills the lungs with air—by drawing the chest-wall up 
and so enlarging the cavity of the chest. 



Fig. 204.—Inducing Artificial Respiration. 

The arms are now swung back to their former position 
alongside the chest, making strong pressure against the lower 
ribs, so as to drive the air out of the chest and to effect an act 
of expiration. This need occupy but a second of time. 

If this plan is regularly carried out, it will make about 
sixteen complete acts of respiration in a minute. It should 
be kept up for a long time, until there is no doubt that the 
heart has ceased to beat or until natural respiration is re-estab¬ 
lished. The cessation of the pulse at the wrist amounts to 
nothing as a sign of death. Often life is present when only a 
most acute and practiced ear can detect the sound of the heart. 

Respiration having been re-established, stimulants should 



86 o 


CARE OF HEALTH . 


be given as soon as they can be swallowed. A teaspoonful 
of whiskey in a tablespoonful of hot water may be given every 
few minutes until the danger point is passed. Warmth must 
be secured immediately by any means available, as hot bot¬ 
tles, plates or bricks, warm blankets and wraps. The body 
must be constantly and effectively rubbed, the direction of 
the rubbing being towards the heart to help the labored cir¬ 
culation of the blood. Meantime every effort should be con¬ 
tinued to restore respiration. No attempt should be made 
to remove the patient, unless he be in danger from cold, until 
the restoration has been thoroughly accomplished. 

382. Serpent- and Insect-bites. —The bites of Poisonous 
Snakes demand instant cauterization or excision of the injured 
part. A handkerchief should be fastened above the wound 
and a stick be passed through it and twisted to prevent the 
poisoned blood from moving towards the trunk and heart. 
It may be well at first to scarify the wound to enable it to 
bleed freely. Some one should then suck it. If practicable, 
the injured part may be soaked in hot water and squeezed to 
draw the blood out after incision. Immediate application of 
ammonia may be of advantage. The safest procedure of all 
is immediate excision of the part, or cauterization with a 
needle heated to redness. 

Among Insect-bites , the most annoying are those of the 
chigre . The treatment must be applied immediately and be¬ 
fore the insect lays its eggs. This consists in anointing the 
bites with a 10 % solution of iodoform in collodion. Where 
the pest abounds, each individual should wear close-fitting 
leggings or top boots, and each day on returning to camp 
should bathe the whole body with salt water. Lime-juice, 
lemon-juice, kerosene oil, or salt pork rubbed over the infected 
parts of the body prevent the chigres from entering the skin 
by removing them. 

383. Surgical Advice. —In cases of Burns or Scalds re¬ 
move immediately with scissors all clothing about the injured 


MED I CINE- CUES T. 


861 


part. Then dress with sweet-oil, castor-oil, or sweet lard, but 
no oil containing salt should be used. Caron-oil, which is a 
mixture of linseed-oil and lime-water, gives the greatest relief. 

In Sprains of all sorts, as those of the wrists or joints, the 
immediate effort should be to rest the tendons by covering 
the parts with cotton wool followed by a soft, firm bandage. 
Next, the inflammation should be allayed by the application 
of hot water; finally, the absorption of inflammatory products 
should be promoted by friction, kneading of the joint, careful 
motion of it, and alternate hot and cold douching. 

Wounds or Clean Cuts should be treated by bringing the 
edges together after washing with antiseptic solution, and 
then supporting them in that position by long strips of 
adhesive plaster. These should not be applied to the wound, 
but first to one side of it and, drawing the flesh together, to 
the other side so as to bring the cut parts in contact. 

Hemorrhage of Vein-blood should be treated by the eleva¬ 
tion of the part and the application of cold water, ice, snow, 
salt, or vinegar. In addition to a severe application of cold, 
firm intense pressure should be applied below the wound, and 
this generally suffices to stop it. 

Arterial Hemorrhage , known by the bright color of the 
blood and its spouting in jets, must be controlled from above , 
i.e., on the side towards the heart, and in the same manner 
as for venous hemorrhage, but by the application of firm 
pressure over the artery, if it can be located—which it fre¬ 
quently can by noticing its pulsations. Stimulants should 
not be given at all, or with the greatest caution, in case of 
hemorrhage, as they excite the circulation of the blood. 

384. Medicine-chest. —No. 1. Tincture of opium: seda¬ 
tive. Dose, 10 to 30 drops in water, not to be repeated for 
six hours. In diarrhea, dysentery, pleurisy, colic, sleepless¬ 
ness, etc. 

No. 2. Paregoric: sedative. Dose, 15 to 60 drops in 
water. In colds, coughs, bronchitis. 


862 


CARE OF HEALTH. 


No. 3. Chlorodyne . Dose, 5 to 25 drops in water. In 
seasickness, diarrhea, colic, cramps, spasms, neuralgia. 

No. 4. Turpentine. For fomentations; to be sprinkled 
on flannels wrung out of boiling water and at once applied to 
the skin. In colic, dysentery, pleurisy, pneumonia. 

No. 5. Carbolic acid. Used in solution and externally 
only: 1 part to 100 parts water to remove foul odors or to 
wash wounds; 1 part to 20 parts olive- or linseed-oil as an 
application to ulcers, to prevent attacks from insects, to de¬ 
stroy ticks, etc. 

No. 6. Olive-oil. For use with above; also as a local 
application to burns, etc. 

No. 7. Opium pills , one grain each. Dose, one pill. In 
diarrhea, rupture, spasms, colic, etc. 

No. 8. Dover s powder , in capsules or tablets of five 
grains. Dose, one to two capsules. In bronchitis, coughs, 
colds, pleurisy, dysentery, fevers, etc. 

No. 9. Calomel and soda , in one-grain compressed tablets. 
Dose, 1 to 5 tablets. In torpid liver, disordered stomach, 
liver congestion, pleurisy, diarrhea, etc. 

No. 10. Quinine , in five-grain capsules. Dose, one to 
five capsules. In malarial fevers, etc. 

No. 11. Ipecacuanha powder, in five-grain capsules. Dose, 
one to six capsules. In dysentery, especially in the premoni¬ 
tory or acute stages; also as an emetic after poisons. 

No. 12. Salicylate of soda (purest procurable), in ten- 
grain capsules. Dose, one to eight capsules a day. For 
rheumatism of all kinds. 

No. 13. Vaseline. For use as simple ointment. 

No. 14. Permanganate of potash , in two-grain pills. For 
snake-bite, internally; as surgical wash or as a gargle for sore 
throat, one dissolved in a cup of water; also for snake-bite 
injected hypodermically close to the wound. 

No 15. Adhesive plaster, tape rolled in tin. 


MED I CINE- CUES T. 863 

No. 16. Mustard , in tin, and vmstard-leaves. For counter¬ 
irritation and as an emetic. 

No. 17. Two clinical thermometers in cases. For certain 
detection of fevers when temperature is noted above 99 0 F. 
This invaluable but fragile instrument should be carried in 
duplicate in case of accident. 

No. 18. Several long cotton roller bandages , various 
widths; a rubber bandage and two pairs of triangular band¬ 
ages for fractures. 

No. 19. Borated lint and absorbent cotton. For dressing 
wounds and sores. 

No. 20. Arrowroot. As a food after fevers and dysentery 
and after violent vomiting. 

No. 21. Persulphite of iron. Applied to wounds to stop 
violent hemorrhage. 

No. 22. Sun cholera tablets. For use in cases of diarrhea, 
cholera, etc. Dose, one every two hours until three or four 
have been taken. 

No. 23. Extract of beef (Liebig). For beef tea and broth. 

No. 24. Collodion with 2 \I<> salicylic acid. For insect- 
stings, skin eruptions, and corns, to be used as a paint. 

No. 25. Collodion with 10 % iodoform. To be painted cn 
wounds as a dressing. 

No. 26. Carbolized vaseline. For dressing wounds. 

No. 27. One hypodermic syringe. 

No. 28. One dozen assorted surgical needles and silk. 

No. 29. Styptic cotton. For nose-bleed at high altitudes. 

No. 30. Iodoform. For dressing wounds and sores. 

No. 31. Vegetable compound cathartic pills. For torpid 
liver and constipation. Dose, one to three pills. 

No. 32. Linseed . For poulticing boils, the abdomen, etc. 

No. 33. Castor-oil in capsules. As a mild laxative or pur¬ 
gative. Dose, one-half to one fluid ounce. 

No. 34. Bichloride of mercury tablets. For antiseptic wash 
for wounds and sores. 


CHAPTER XLI. 


PHOTOGRAPHY. 

385. Uses of Photography in Surveying. —As a map 

record alone is insufficient to completely illustrate the results 
of an exploratory survey, requiring for the fuller understand¬ 
ing of the discoveries made a written report as an accompani¬ 
ment, so also is such a report incomplete unless accompanied 
by illustrations (Chap. IV). A military reconnaissance must 
likewise be accompanied by a report, and this is made more 
comprehensive, and is often more rapidly and lucidly prepared, 
when illustrated by sketches or photographs (Chap. V). 

The present stage reached in the development of the sci¬ 
ence of photography is such that any one possessing the qual¬ 
ifications necessary for the execution of an exploratory, 
geographic, or military reconnaissance could easily acquire the 
skill necessary to make photographs for the proper illus¬ 
tration of the accompanying report. The varieties of work 
to be executed under such circumstances are many. They 
include chiefly outdoor or landscape photography, but in ad¬ 
dition must frequently be accompanied by illustrations of the 
inhabitants, the fauna and flora, as well as details of the geol¬ 
ogy of the regions traversed. Finally, photography is now 
employed quite extensively in the making of topographic sur¬ 
veys (Chap. XIV), and may be used in determining longi¬ 
tudes (Chap. XXXVII). 


864 


CAMERAS. 865 

386. Cameras. —There are two general types of photo¬ 
graphic cameras: 

1. The hand camera; 

2. The tripod camera. 

The former are of three general kinds, namely: 

a. I hose with lenses having universal focus. 

b d hose which require to be focused by means of a scale 
attached to the camera; and 

c. Those which, in addition to being used as hand cam¬ 
eras, may be mounted on tripods and focused as are stand 
cameras. 

The stand camera is mounted upon a tripod by which 
it must always be leveled when used. It is provided with a 
ground glass on which the image sighted can be focused by 
means of a ratchet motion and the bellows attachment of the 
lens. For general exploratory uses the extension bellows 
should be of red Russia leather, as red ants will not eat this. 
According as the rear or front end of the camera is moved by 
the rack motion the camera is said to have a front or back 
focus. In addition it should be possible to move the lens 
vertically through a short space in order to take in images 
which cannot be reached without tilting the camera out of 
level, which should never be done. The lens and the plate- 
holder should both tilt a little to aid in seeing objects with¬ 
out the range of view. 

Hand cameras are made in all sizes, from those carried in 
the pocket, which take pictures 2 \ by 3J inches, up to those 
which take pictures 5 by 7 inches and are provided with 
all of the refinements which permit of doing the best tripod 
work. Hand cameras, however, are rarely provided with 
a sufficiently fine lens for the highest grade of photographic 
landscape work. For this reason and because the larger hand 
cameras have to be focused and are therefore not so handv 
nor so satisfactory either in manipulation or results as the 
smaller hand cameras of universal focus, the best camera out- 


866 


PHOTOGRAPHY . 


fit for the explorer would be a small fixed-focus hand camera 
and a first-class tripod camera. Such a tripod camera should 
be capable of taking pictures either of 5 by 7 or of 6J by 8^ 
inches, and should be provided with the best combination 
wide and narrow angle lenses procurable. 

For photographic surveying the relative positions of plate 
and lens in the camera must be invariable, and when adjusted 
the plate must be exactly vertical. Accordingly but few of 
the makers are able to supply suitable cameras where the work 
is to be accurately executed. Nearly any camera may be 
readily adapted to the work of reconnaissance surveying by 
photographic methods when there is oriented upon it a com¬ 
pass for directions and cross-hairs or needle pointers to fix ini¬ 
tial directions. 

A scale showing the position of the lens for focusing on 
objects at various distances should be applied to all stand 
cameras in the same manner as it is applied to high-grade 
hand cameras. All good tripod cameras should in addition 
have the position of the universal focus and the corresponding 
stop number, as F 16, etc., marked on the focusing-rack. 
Where the camera is provided with ground glass and oppor¬ 
tunity permits, this should always be used in focusing. 

The tripod camera requires a focusing-cloth to be thrown 
over the head of the operator when focusing on the ground 
glass, thus producing for him a small dark room in which to 
observe the glass. This focusing-cloth should be of rubber or 
of stout black baize or similar material, having a quantity of 
leaden shot sewn in a hem around the edges to prevent its 
being blown about in the wind. An additional ground glass 
should always be carried in the field as a precaution against 
breakage. Celluloid covered with ground glass substitute 
is useful in case of breakage. If a little powdered emery 
be carried, a glass plate may be cleaned off and the same 
quickly turned into ground glass by rubbing with emery and 
cloth. 


LENSES AND THEIR ACCESSORIES. 


867 


387. Lenses and their Accessories. —For the ordinary 
purposes of photography there are three classes of lenses, 
namely: 

1. Portrait lenses; 

2. Landscape lenses; 

3. Copying lenses. 

Portrait lenses require a large aperture compared with the 
. focal length, so as to admit a large volume of light in the sub¬ 
dued atmosphere of a room. They are aplanatic; that is, 
they can be used without a stop. They have little depth of 
focus, narrow field, and great rapidity. 

Copying lenses must be achromatic and anastigmatic. They 
should be rapid and rectilinear. 

The type of lens best suited to the purposes of the ex¬ 
plorer or surveyor is that which may be used for general 
view-work, where great flatness of field is unnecessary and 
distortion must be a minimum. 

There are two general classes of landscape lenses , namely: 

1. Single achromatic lenses; 

2. Combination lenses. 

The single lens is that most used in instantaneous hand 
cameras, and is of the achromatic converging-meniscus type. 
The flatter the lens the more rapid. The defects of such lenses 
are, distortion of the image, moderate angular view and slow¬ 
ness, but when used with small stops they have depth of focus 
and produce crisp negatives. They are nonaplanatic and must 
be used with stops. The defects of the single landscape lens 
are largely corrected by the combination lens, which possesses 
many of the best qualities of the copying lens. Combination 
wide and narrow angle lenses , now made for the best land¬ 
scape work, will take wide-angle views with the two glasses, 
and narrow-angle views with one glass removed. 

Distortion as applied to lenses is due to the greater refrac¬ 
tion of the rays from the margin of the lens towards the axis. 
This defect is most pronounced in single lenses without dia- 


868 


PHOTOGRAPHY . 


phragms. Aberration is due to the impossibility of obtaining a 
good definition in attempting to focus on the ground glass. 
The image appears as a circular patch of light, decreasing in 
intensity from the center to the edges. It increases as the 
square of the aperture of the lenses, and inversely as its focal 
length. It is of two kinds: I. Lateral; and 2. Longitudinal. 
As aberration increases so rapidly with the aperture, stops or 
diaphragms are employed to reduce it. Their effect is to cut 
off the marginal rays. In use the camera is focused with a 
large diaphragm, and a smaller one is employed in making the 
exposure. In a single lens diaphragms are placed in front. 
In combination lenses they are placed between the lenses. As 
the brightness of the image depends upon the quantity of 
light admitted by the diaphragm, it is proportioned to its 
aperture or the square of its diameter. The larger the aper¬ 
ture the more light admitted. Brightness further varies in¬ 
versely as the square of the focal length. Thus by doubling 
the focal length the dimensions of the image are doubled and 
the light admitted is distributed over four times the area. 
The brightness of the image is reduced in proportion. 

For the highest type of photographic surveying the lens 
must be (i) rectilinear; (2) free from distortion; (5) it should 
cover an angular field of about sixty degrees, and (4) the 
definition should be uniform over the entire plate. Slowness 
is preferable to rapidity in order to get strength of shades and 
definition. For these purposes what are known as wide-angle 
lenses must be employed. They are doublets of two lenses 
between which is placed the diaphragm. For photographic 
surveying on the Canadian Government a Zeiss anastigmatic 
of F 18 aperture and 14-millimeter focus is preferred. Such 
doublets consist of an achromatic interior in which the flint has 
the higher refractive index. Among rapid rectilinear anas¬ 
tigmatic combinations are those made from Jena glass. 

When attempting panoramic view-work from the summit 
of a mountain it becomes occasionally desirable to take a 


DRY PLATES AND FILMS . 


869 


more detailed view of some object in the field. This cannot 
be done without approaching more nearly to the object with 
the camera. There is, however, an attachment called a tele¬ 
photo combination which consists in the addition of a negative 
or magnifying element in the rear of the combination proper. 
This produces larger images of distant objects, but it must 
be remembered that by reducing the light it necessarily 
reduces the rapidity of the combination lens. 

The lenses supplied with cameras have, as a part of their 
construction, some kind of shutter for making both time 
and instantaneous exposures. These are operated by cylin¬ 
ders in which a piston is actuated by compressed air from a 
hand-bulb. The pressure on this bulb opens and closes the 
shutter. The best shutters are so automatic in their con¬ 
struction that by setting a pointer to the required speed in 
seconds or fractions of a second, as marked upon it, the 
proper time is given by the shutter upon applying pressure to 
the bulb, or it sets the shutter in the time exposures. 

388. Dry Plates and Films. —The effort in all good 
photographic work is to obtain results full of detail and 
clearly defined. The most difficult operations in photography 
are the procuring of clearly defined landscape views because 
of the character of the light diffused by the atmosphere. As 
a result aerial perspective is much exaggerated as produced 
by photography, because of the strong actinic effect of the 
blue haze through which distance is seen and which speedily 
blurs out the details of the image. The presence of smoke 
or dust in the air contributes to the same result. 

This effect can be eliminated to a certain extent by selec¬ 
tion of dry plates especially suited to such work. Ordinary 
plates are sensitive to the blue and violet rays only. Ortho- 
chromatic or isochromatic plates are manufactured which are 
acted upon by the colors at the other end of the spectrum, 
although the maximum sensitiveness is still to blue and 
violet. By using a screen or ray filter of orange or reddish- 


870 


PHOTOGRAPHY. 


orange glass it is possible to exclude the greater volume of 
the light rays other than the green, yellow, and red, and 
such a screen in connection with the orthochromatic plates 
partly solves the difficulties of photographing through haze. 
All that is required in such a plate is that it be especially 
sensitive to other rays than blue and violet, because these are 
largely cut off by the screen. 

The proportion between direct sunlight and skylight varies 
with the altitude of the sun and with the absorption of the 
atmosphere. Shadows look more intense when the sun is 
high than when it is low. Accordingly, on mountains and 
in general landscape view-work at high elevations the contrast 
is greatest because the atmosphere is very light and the 
coefficient of absorption proportionately small. For this 
reason good photographs of mountain scenery are scarce, and 
also because of the wide contrast ranging from snow and sun¬ 
light to dark woods and shade. In such work satisfactory 
results can only be expected when orthochromatic plates or 
color screens are used. 

When a subject presenting strong contrast is given long 
exposure to the action of the light, the image appears to 
spread upon the plate. The edges of the high lights merge 
into the shadows by a gradually decreasing tint, according to 
the intensity of the light and the length of the exposure. 
This is called halation and is due to the light which has 
passed through the film, striking the rear surface of the 
glass plate and being reflected by it on the back of the film, 
causing there a halo. The remedy of halation is to stop the 
light when it reaches the back surface of the plate by coating 
the latter with some non-actinic material which will absorb 
light. Any kind of opaque material will not do. The coat¬ 
ing must be in optical contact with the glass, and the refrac¬ 
tive index of the coating must be the same as that of the 
glass. Such a coating is produced by painting the back of 
the glass with a solution of fine lampblack mixed in sandarac 


DRY PLATES AND FILMS 


S 7 I 


dissolved in alcohol. Nonhalation produced by such a coat¬ 
ing is rarely necessary in ordinary photography, providing 
one uses dry plates which are orthochromatic . Where, how¬ 
ever, it is necessary to photograph towards light, as viewing 
in the direction of the sun or towards electric lights at night 
or their reflections, even the best grades of orthochromatic 
nonhalation plate may be reinforced by the aid of the non¬ 
halation backing. Before developing the plate so backed, 
the lampblack must be washed off with alcohol. 

The isochromatic nonhalation plates furnished by the deal¬ 
ers have been made nonhalation by coating them with two or 
three layers of the silver emulsion. Thus the Seed nonhala¬ 
tion plates are given a coat of emulsion of 23 sensitometer 
test followed by a second coat of emulsion of 26 sensitometer 
test. Wuerstner triple-coated isochromatic nonhalation plates 
have three coats numbered 1, 2, and 3. The principle on 
which these work is that the light coming from the blue and 
the violet rays makes its way more quickly through the first 
coating, and is impressed upon the second or third coating at 
about the same time that the light coming from the less rapid 
rays reaches the first or second coating. 

Where the best work is attempted in photographing distant 
views, dry plates prepared as above must be used. For the 
general purposes of the photographer, explorer, or military 
photographer, however, where the effort is merely to procure 
a good record of objects seen, the most satisfactory plates 
are the simple single emulsion plates without isochromatic or 
nonhalation character. These may be of glass or on cut celln- 
'aid films. The latter give excellent results for all practical 
purposes of the amateur photographer. They will not break 
in transportation, are less heavy and bulky than glass, and 
are in every way more satisfactory where compactness of 
outfit is an item. For use in the instantaneous hand camera 
roll films should be used. With the larger sizes of hand 
cameras it is not possible to stretch the film sufficiently, and 


872 


PHOTOGRAPHY. 


parts of it are thus out of focus, due to wrinkles in it. The 
best cut films are now prepared with isochromatic nonhalation 
coatings, and give results almost equal to the best glass dry 
plates. 

389. Exposures. —The best results in exposing plates in 
photographic work are to be procured by using slow plates. 
This gives sufficient time to bring out the deep shadows, and 
there is less likelihood of error in properly timing with slow 
than with fast plates. Where the hand camera is used in¬ 
stantaneous exposures should be made whenever the light will 
permit, the best results only being had with such a camera by 
rapid work. Where landscape work or panoramic work from 
an eminence is to be done, or where detail of a strongly marked 
object is to be brought out, the tripod camera and slow ex¬ 
posure should be used. 

For a plate having a speed which, under ordinary con¬ 
ditions, requires one-fifth of a second for exposure, a small 
fraction of a second overtiming or undertiming affects the plate 
more seriously than in the timing of a five-second plate. In 
the latter case a fraction of a second under timing or overtim¬ 
ing is of small moment. In exposing a slow plate it is best 
always to err in the direction of overtiming, a little more time 
doing less harm than undertiming. 

The exposure to be given a plate is inversely proportional 
to the intensity of the light illuminating the object. A sub¬ 
ject requiring an exposure of ten seconds with the intensity 
of light taken as one, will require an exposure of five seconds 
with an intensity of two. The light received by a landscape 
in direct sunshine consists of: 1, Direct rays of the sun; 2, 
The light diffused by the sky. As a result there is considera¬ 
ble change in the exposures required at the same time of day 
at sea-level and at great altitudes. It has been found that 
there is little change in the exposures required at great eleva¬ 
tions until the sun approaches the horizon. According to Mr. 
E. Deville, taking the exposure with the sun at the zenith as 


EXPOSURES. 


873 


one second at sea-level, the exposure at 10,000 feet altitude 
will be a trifle under one second. With the sun at 40 degrees 
altitude at sea-level, one and one-fourth seconds will be re¬ 
ps 
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quired, whereas at 10,000 feet altitude one second will still 
be sufficient. As the altitude of the sun decreases the differ¬ 
ence is rapidly accentuated. At 25 degrees altitude one 
second is required at 10,000 feet, two seconds at sea-level; 































































































874 


PHOTOGRAPHY. 


at 15 degrees altitude one and one-fourth seconds at 10,000 
feet, three and one-half seconds at sea-level. 

Fig. 205, from Lieut. Reber’s “ Manual of Photogra¬ 
phy,” shows how the photographic intensity of daylight varies 
with the time of day and of year. Table LXXII, also taken 


Table LXXII. 

RELATIVE TIMES OF EXPOSURE FOR DIFFERENT STOPS AND 

SUBJECTS. 


Stop. 

Open 

Landscape. 

Landscape with 
Heavy Fol’age 
in Foreground. 

Under Trees. 

Fairly Lighted 
Interiors. 

Badly 

Lighted 

Interiors. 

F 8 

O.25 

1.50 

150 

150 

1440 

F 11 

O.50 

3.00 

300 

300 

2880 

F 16 

1.00 

6.00 

600 

600 

5760 

F 32 

4.OO 

24.OO 

24OO 

2400 

23040 

F 64 

16.00 

96.OO 

9600 

9600 

92160 


from Reber, shows the times of exposure required with vari¬ 
ous sizes of stop in different subjects. This is arranged on 
the basis of the Carbutt B 16 plate with F 16 stop and normal 
exposure on open landscape on a day when three seconds 
of time is sufficient. Such an exposure is taken as unity. 
Accordingly, opposite F 8 in the table, under badly lighted 
interiors, the exposure 1440 should be multiplied by 3 sec¬ 
onds, the time given the unit Carbutt B 16 plate. As a result 
it is found that the time required under these changed condi¬ 
tions will be 1 hour 28J minutes. If, in experimenting with 
another and faster plate, as Carbutt Eclipse 27, one-fourth of 
a second is required, the proper exposure with the F 64 stop 
on the landscape with foliage would be 96 multiplied by J sec¬ 
ond, or 24 seconds. Thus by reference to this table and by 
making one experimental exposure the operator may know 
what exposure to give under different conditions. He must 
also keep in mind, however, the time of day and season of the 














DEVELOPING . 


875 


year as related to that at which the experimental exposure 
was made. Then by reference to the diagram he will have a 
fair idea of the time required. 

Referring now to the diagram (Fig. 205), if the three-sec¬ 
ond exposure taken above as unity were made at 5 P.M. in 
March or August, the same subject and plates would require 
but two seconds at 5 P.M. in June. It would require 60 
seconds at the same hour in November or at 7 a.m. in the 
months of March, June, or August. Again, an exposure 
made at 7 P.M. in June would require only as much time as 
one made at 5 P.M. in November. 

Much depends upon the coloring and briltiancy of the ob¬ 
ject, especially whether there is much green, yellow, brown, or 
red in it. Distinct views require less time than less clearly 
defined ones. Exposures are often made only for distance, 
others for foreground only. For deep shadows long exposure 
is required. 

390. Developing —Before developing a plate it should be 
cleaned with a camel’s-hair brush to remove dust from the 
surface which would otherwise produce pin-holes in the nega¬ 
tive. The plate should also have been dusted before placing 
it in the plate-holder before exposure. Where an effort is 
made to obtain the best results, only filtered or boiled water 
or water from melted ice should be used. The developing 
solution should be at a temperature of from 65 to 70 degrees, 
and must never be warmer than this. When water is warm 
enough to cause the emulsion to frill after development, the 
plates will be greatly helped by first flowing over the surface, 
previously wetted with water, a strong solution of alum water . 
The negative should be handled only by the edges, great care 
being taken not to touch the film side. 

Before placing in the developer the plate should have water 
flowed over it to thoroughly wet it, and should be placed 
emulsion side up in the developing tray. The developer should 
then be flowed quickly back and forth over the surface of the 


876 


PHOTOGRAPHY. 


film by a sweeping movement, so that no air-bubbles shall col¬ 
lect on the surface. The wet surface assists the developer in 
spreading evenly. The rocking motion given the tray should 
be continued throughout the process of development. In 
developing roll films these should be laid in the tray and then 
held under the faucet and the surface of the film be swabbed 
with a moist camel’s-hair brush or a bunch of absorbent cot¬ 
ton. The tray should then be drained and the developer be 
applied. This is not necessary, however, with cut films. 

The high lights in the negative should begin to show in 20 
seconds. If they appear too quickly, the plate has been over¬ 
exposed and the action of the developer should at once be 
checked. For this purpose, pour off the developer, and if 
necessary wash the plate with a little water. Then begin 
again with a developer to which has been added 10 drops of a 
10 per cent solution of a potassium bromide. If, on the other 
hand, the image does not appear in 30 seconds, the plate is 
probably under-exposed. It is more difficult to bring out the 
detail in an under-exposed plate than to check the develop¬ 
ment of an over-exposed plate. The details in the under¬ 
exposed plate maybe helped by the use of a little more alkali, 
and the density of the negative may be helped by more reducing 
agent. Long soaking in weak or old developer is the best treat¬ 
ment. When the high lights develop properly and then 
thicken before the details come out in the remainder of the 
plate, the developer should be diluted with two or three times 
its volume in water, or some bromide solution be added, thus 
permitting the development to proceed slowly. It may 
also be necessary to leave the plate in a weakened developer 
for a half-hour in order to properly bring out the shadows. 

Developing solutions can be purchased of all dealers. For 
easy and safe transportation they are now put up in tablet 
form by Wyeth & Co. of Philadelphia. If one desires to 
prepare his own developer, the following are recommended : 

The more common developer, and perhaps the most valu- 


DEVELOPING. 


8 77 


able, is the pyrogallic acid solution. This will not keep well, 
however, excepting in well-corked bottles, and when old 
stains the negative yellow. For the best work fresh devel¬ 
oper should be mixed for every two or three plates, accord¬ 
ing to the following formula: 

Crystallized sulphite of soda.120 grains 

Crystallized carbonate of soda.60 “ 

(Or Dry granular carbonate of soda. 30 “) 

Carbonate of potassium. 30 “ 

Then add 10 grains of Schering’s or Merck’s pyrogallic acid 
and 10 minims of a 10 per cent solution of bromide of potas¬ 
sium. 

A pyro solution which will keep for a long time fairly well 
is the following: 


PYRO DEVELOPER. 


Distilled or ice water. . 10 ounces 

Oxalic acid. 15 grains 

(Or, Sulphuric acid. 15 minims) 

Bromide of potassium. 30 grains 


Then add one ounce of Schering’s or Merck’s pyrogallic 
acid and enough water to make 16 fluid ounces. 

An alkali solution for helping out the details of tinder- 
exposed negatives is the following: 

ALKALI SOLUTION. 

Distilled water. 16 ounces 

Sulphite of sodium, crystals. 4 

Carbonate of sodium, crystals. 2 

Carbonate of potassium. 1 

A developer which keeps better than the pyro and is less 
liable to stain when old is the following: 














878 


PHOTOGRAPHY . 


EIKO CUM HYDRO DEVELOPER. 

SOLUTION NO. I. 

Distilled water. 32 ounces 

Sulphite of sodium, crystals. 4 “ 

Eiconogen. 330 grains 

Hydrochinon .. 160 “ 

solution no. 2 . 

Distilled water. 32 ounces 

Carbonate of soda, crystals. 2 “ 

Carbonate of potassium. 2 “ 

For instantaneous exposures take 4 ounces of water, 1 
ounce of No. 1 and 1 ounce of No. 2; for normal exposures 
on rapid plates, 3 ounces of water, 1 ounce of No. 1 and £ 
ounce of No. 2; for normal exposures on slow plates, 4 
ounces of water, 1 ounce of No. 1 and j ounce of No. 2. 

391. Fixing. —After the negative has been developed it 
must be washed, preferably under the faucet, until all the de¬ 
veloper has been removed. It is then placed immediately in 
a clearing and fixing bath, which may be prepared as follows: 


SOLUTION A. 

Warm distilled water. 48 ounces 

Hyposulphite of soda. 16 “ 

SOLUTION B. 

Crystallized sulphite of soda. 2 ounces 

Warm distilled water. 6 “ 


Add 1 dram of sulphuric acid to 2 ounces of water and 
pour into B solution. This latter mixture should then be 
added to the hyposulphite or A solution. Before using the 
fixing-bath dissolve 1 ounce of chrome alum to 8 ounces of 
water and add this to it. This fixer will last a long while 
and may be used over and over. It both clears and hardens 
the negative. An inferior fixing-bath consists simply of hypo¬ 
sulphite of soda dissolved in 4 parts of water. 













FIXING. 


8; 9 


The negative should be left in the fixer about five minutes 
after the white milky bromide of silver has entirely disap¬ 
peared from the film. Then it should be washed for three- 
quarters of an hour in running water and placed in a rack to 
dry. If for any reason rapid drying is necessary, this may be 
accomplished by flowing methyl alcohol or wood alcohol 
over the negative two or three times, which will take up the 
water. In hot weather a bath consisting of a solution of alum 
in water should be used both before and after fixing. If the 
negative is not sufficiently dense for printing, it should, after 
thoroughly washing the last traces of hyposulphite, be placed 
in an intensifying solution. 

A good intensifier is a weak solution of equal parts of mer¬ 
curic chloride (corrosive sublimate) and chloride of ammonium, 
and this should be flowed over the plate until its surface is 
slightly chalky; the longer the solutionis used the denser 
will the plate become. Afterwards it should be washed with 
water, then with a weak solution of chloride of ammonium 
and, after being thoroughly washed, immersed in a bath of 
io minims of strong ammonia to each ounce of water until 
the plate blackens throughout, when it should be washed and 
dried in a rack. To diminish the density of an overdeveloped 
negative, treat the plate with one part of saturated solution 
of potassium ferricyanide mixed with ten parts of io per cent 
solution of hyposulphite of soda. 

The following tabular arrangement from Reber is an index 
to the various causes of defects in negatives: 

Defects in Negatives. Cause. 

I. Fog. Over-exposure ; white light entering 

camera or dark room ; unsafe de¬ 
veloping light ; old and decomposed 
developer ; silver nitrate or hypo¬ 
sulphite of soda in developer ; de¬ 
veloper too warm ; too much alkali 
and not enough bromide in de¬ 
veloper. 


88 o 


PHOTOGRAPHY. 


Cause. 

Under-development. 

Under-exposure. 

Over-exposure and incorrect de_ 
velopment. 

Developer too strong or too warm, 
or too long applied. 

Using too stiff a brush in dusting 
plates, or slide of plate-holder rubs 
against the surface of the plates or 
films. 

Air-bubbles on plate during develop¬ 
ment, or defects in emulsion. 

Dust or muddy water. 

Old developer or washing insuffi¬ 
cient to eliminate hypo. 
Precipitation from old hyposulphite 
bath containing alum. 

Imperfect elimination of hypo. 
Reflection into emulsion by the glass 
back of the light transmitted 
through emulsion. May be pre¬ 
vented by coating the back of neg¬ 
ative with a black wash, or by using 
an emulsion of such thickness as to 
absorb all light falling on it. 

392. Printing and Toning. —Various papers for printing 
can be procured of photographic-supply dealers. Special 
papers, as bromide paper for dead-black prints or platinum 
paper for black or sepia prints, are accompanied by full descrip¬ 
tions for printing. The common photographic print is made 
on silver or albumen paper, and also can be obtained more 
cheaply and satisfactorily from the manufacturers than it can 
be made, and it also is accompanied by descriptions of the 
method of printing. 

It is better to varnish the negative before printing in order 
to prevent scratching or otherwise injuring the film. The 
following is a good, tough, hard and durable varnish: 


Defects in Negatives. 

2. Weak negatives with clear 

shadows. 

3. Strong with clear shadows. 

4. Weak negative with details well 

out in shadows. 

5. Too much density. 

6. Fine transparent lines. 


7. Round transparent spots. 

8. Pin-holes. 

9. Yellow stains. 

10. Mottled negatives. 

11. Crystallization on negative. 

12. Halation. 


PRINTING AND TONING. 


881 


NEGATIVE VARNISH. 


Shellac. 

Mastic. 

Oil of turpentine . 

Sandarac. 

Venice turpentine 

Camphor. 

Alcohol.. 


ij ounces 

X. < i 

4 

i “ 

o l “ 

1 << 

4 

20 grains 
20 ounce 


To print , the negative is placed in the printing-frame, film 
side up, and back of this is placed a piece of sensitized paper 
of the same size as the negative, with the silver side down or 
facing the negative. The whole is backed by blotting-paper 
to fill the frame, which is at once closed and stood on edge in 
such manner as to expose it to the direct rays of the sun. 
The printing should continue until the shadows bronze out 
well, the operator keeping in mind that the print will be less 
strong when toned and fixed. In examining the print this 
should be done only in subdued light, great care being taken 
to raise only one-half of the back at a time and not to let the 
negative or paper slip. Where the best class of work is at¬ 
tempted the printing should not be done under the direct rays 
of the sun, but under ground glass on a cloudy day, or in 
subdued light, thus procuring softer results. 

After removing the prints from the frames they should be 
kept in a dark box until toned, which should be done within 
24 hours. Prints are not fixed immediately after printing 
because of the disagreeable reddish color produced. To o'» 
viate this, where pleasing effects are desired, the print is first 
toned by placing it in a solution of chloride of gold, the suit 
of which comes in very small sealed bottles and is best k pt 
by dissolving it in water in proportion of one grain to tiie 
ounce. 

To get good purple and black colors immerse in a ton; • - 









882 


PHOTOGRAPHY. 


bath of 9 ounces of water to i ounce of gold solution 
neutralized by a little carbonate of sodium until the bath is 
alkaline, as shown by the testing-paper. A rich warm tone 
recommended by Reber is i ounce of gold solution to 30 grains 
of acetate of sodium in 8 ounces of water. The longer the 
prints remain in this bath the browner the tone. The most 
satisfactory results are procured by preparing the bath at least 
a day before using. 

Before toning prints they should be washed face downward 
by laying them in a basin of water until there is no trace of 
cloudiness in the water, each print floating separately by 
itself. The prints should then be laid face up in the to?iing~ 
bath separately, the separation being produced by dropping 
the prints one at a time so that there shall be a layer of liquid 
between them. The tray should then be given a gentle rock¬ 
ing motion until the toning has progressed far enough, when 
the prints should be removed and placed face downward in 
the water to stop the action of the toning solution. A little 
salt added to the water stops the action more quickly and pre¬ 
vents the tendency to blister. 

After toning and washing , the prints should be put in the 
fixing-bath of I part of hyposulphite of soda to 4 parts of 
water, in which they should be left for 15 minutes. When 
fixed the whites will appear colorless and the shadows be free 
from red spots. The fixing-bath will be improved by a dram 
of ammonia added to each 10 ounces of water, as the am¬ 
monia increases the speed of fixing and prevents blistering. 
After fixing , the prints should be washed in running water 
for several hours to remove every trace of hyposulphite, 
which would otherwise cause the prints to lose brilliancy and 
fade. There can now be procured of all dealers a self-toning 
paper which has only to be put in water and hypo. This 
paper is especially convenient in getting proofs in the field. 

The following table is given by Reber as indicating th^ 
defects in prints and their causes: 


BLUE-PRINTS AND BLm CK-PRINTS, 


883 


1. 


2. 



4 - 

5 . 


6 . 


7 - 

8 . 

9 - 

10. 

11. 


12. 


13. 

14. 

15. 

16. 


Defects in Prints. 

Small white spots with black 
center. 

Gray starlike spots. 

Bronze lines, if straight. 

Bronze lines, curved. 

Marbled appearance of print. 

Note.— 3, 4, and 5 refer 


Causes. 

Dust on paper. 

Inorganic matter in paper. 

Line of stoppage during floating of 
paper. 

Scum on sensitizing-bath. 

Baths too weak or not floated enough. 


especially to albumen paper. 


Red spots on prints, especially 
in shadows. 

Weak prints. 

Harsh prints. 

Too red a tone. 

Cold blue tone. 

Streaky prints. 

Whites appear yellow. 

Yellow spots when dry. 

Prints refuse to tone. 

Dark, mottled appearance in 
body of paper. 

Blisters. 


Marks caused by moist fingers com¬ 
ing in contact with paper. 

Weak negatives. 

Harsh negatives. 

Undertoning. 

Overtoning. 

Acid toning-bath. 

Imperfect washing ; imperfect ton¬ 
ing ; not long enough fixing. 

Imperfect elimination of hypo. 

Gold exhausted from toning-baths, or 
there is hypo in separate toning- 
baths. 

Improper fixing in too strong a light. 

Saline solution between emulsion and 
paper. Can be prevented by salt¬ 
ing the first wash-water. 


393. Blue-prints and Black-prints —Both blue-and black- 
print paper can be purchased of dealers in draughting and 
photographic supplies. Blue-print paper can be made readily, 
however, by floating close-grained drawing-paper in a bath of 
1 part of ammoniocitrate of iron and 2 parts of water,to which 
is added I part of ferricyanide of potassium dissolved in 4 
parts of water. This bath must be kept in the dark and 
used immediately. The paper may then be removed and 
thoroughly saturated and hung up to dry by spring clothes¬ 
pins. Blue-prints are printed by placing the tracing over the 
blue-print paper and exposing it to direct sunlight. Print¬ 
ing should continue until the surface is well bronzed or blue, 


884 


PHOTOGRAPHY. 


according to the paper used, when it is at once placed in an 
abundance of flowing water and washed until free from all 
blue or yellow. The blue-prints are then either hung up to 
dry or, better still, placed between blotting-paper. 

Black-print paper, or that which produces black lines oil- 
white paper, may be prepared by immersing drawing-paper in 


the following solution: 

Water. 9 ounces 

Gelatin. 3 drams 

Solution of perchloride of iron (U. S. 

Ph.). 6 drams 

Tartaric acid. 3 drams 

Ferric sulphate. 3 drams 


After printing develop in the following solution: 

Gallic acid. 6 drams 

Alcohol. 6J ounces 

Water. 32 ounces 

Then wash and thoroughly dry. 

Another formula for the same is that given by Mr. B. 
Howarth Thwaite, and is as follows: 

1. Gum arabic. 12 drams 

Water. 17 ounces 

2. Tartaric acid. 13 drams 

Water. 6 ounces 

3. Persulphite of iron. 8 drams 

Water.6 ounces 6 drams 

The paper to be prepared by immersion, separately in 
1 and 2, and to be developed in 3. 
















INDEX. 


A, B, C star factors, 758 
Aberration, correction for diurnal, 
754 

in lenses, 868 

Abney clinometer hand-level, 357 
Accidental errors, 580 
Accuracy of astronomic positions, 
744 

of base measurement, 516 
of primary traverse, 536 
and probable error of primary tri¬ 
angulation, 553-59 2 
Acre, English equivalents of, 673 
Adjusted sketch sheet, 27 
Adjustment of Burt solar, 782 
figure, 616 

general conditions applied to tri¬ 
angulation, 616 
of group of level circuits, 345 
local conditions applied to trian¬ 
gulation, 616 

notation used in figure, 618 
of primary traverse to astronomic 
positions, 542 
routine of figure, 617 
routine of station, 612 
of Smith meridian attachment, 786 
station, 610 
of transit, 204 
of traverse, 202 

Adjustments of camera for photo¬ 
graphic longitudes, 794 
of level, 308 

of telescopic alidade, 159 
Aerial erosion, 111 
Agonic lines, 222 
Airv, Wilfrid, 490 
Alidade, 49 

adjustments of telescopic, 159 
sight, 36 

solar attachment to telescopic, 791 
traversing with, 792 


Alidade, manipulation of straignt=> 
edge of, 178 
Alidades, sight, 161 
telescopic, 21, 157 
Alkali solution in developing, 877 
Altitude, differences of,from vertical 
angulation, 364 

latitude by a measured, 723, 725 
American Ephemeris, 741, 747, 789 
Amphitheater, 131 
Andrews, Horace, 335 
Aneroid, 22, 28, 29, 48, 70, 77, 87, 
100, 395 
errors of, 395 

sketching contours with, 399 
using the, 396 
Angle equations, 619 

and side equations, solution of, 
625 

corrected spherical, 633 
independent measure of, 586 
measuring by directions, 584 
measuring by repetitions, 584 
measurement of, by series, 586 
methods of measuring horizontal, 

584 

number of repetitions of, 585 
sets of measures of, 587 
Angles, plane, 638 

precautions in measuring horizon¬ 
tal, 577 
spherical, 638 
vertical, 29, 49 
Angular tachymetry, 272 
Angulation, vertical, 39, 361 
Animals, forage for, 849 
for packing, 837 
Annual magnetic change, 222 
Aparejo for packing, 838 
Aperture of lenses, 868 
Apparent day, 682 

displacement of celestial body, 6S8 

885 




886 


INDEX. 


Apparent day, motion of heavenly 
bodies, 680 

Aqueous agencies, 111 
erosion, III 

Arc and time, interconversion of, 

698 

Areas of quadrilaterals of earth’s 
surface, 436 

Arithmetic mean, probable error of, 

6c 7 

Army pack-saddle, 839 
Army wall tent, specifications for, 
by Quartermasters’ Department 
of U. S., 820 

Art of topographic sketching, 40 
Artificial respiration, 859 
Astronomer, field-work of geodetic, 
679 

Astronomic checks on triangulation, 
496 

day, 683 

determinations, accuracy of, 497 

formulas, fundamental, 684 

latitude and longitude, 679 

methods, 678 

notation, 6S3 

planes of reference, 681 

position, 76 

position, adjustment of primary 
traverse to, 542 

position, cost, speed, and accuracy 
of, 744 

position, reduction of primary 
traverse to, 527 
stations, 34 

terms, definitions of, 679 
transit and zenith telescope, 726 
Astronomy, Chauvenet’s, 678 
Doolittle’s, 678 
geodetic, 679 
Hayford’s Geodetic, 678 
works of reference on, 809 
Atmosphere, boiling of, 348 
Atmospheric pressure, 374 
Attraction, local, 775 
local magnetic, 223 
Austria, scale and cost of survey of, 75 
Automatic heliotropes, 7 
Automatic surveying instruments, 291, 
Autumnal equinox, 681 [291a 

Azimuth, 146 

approximate solar, 708 

computation of geodetic, 642 

correction for deviation in, 753 

determination of, 707 

at elongation, 721 

errors, in primary traverse, 528 


Azimuth and latitude, determination 
of, with solar attachment, 789 
of a line, 636 
mark, 707 

observations, reduction of, 720 
observing for, 707 
on primary traverse, 535 
zero of, 212, 636 
Azimuths, 76 

correction; for check in primary 
traverse, 539 

factors for computing geodetic, 
649 

of Polaris, 714, 716 
primary, 719 

of secondary accuracy, 712 

Baden, cost and scale of survey of, 67 
survey of, 75 
Bagnall, Gerald, 490 
Bags, sleeping, 850 
Baker, Ira O., 259, 264, 279, 490 
on leveling, 326 
Baker, tin reflecting, 832 
Baldwin, H. L.,503, 792 
record of base measurement by, 
518 

record of triangulation observa¬ 
tions by, 589 

Baltimore, survey of, 64, 67, 104, 107 
topographic survey of city of, 490 
Barnard. E. C., 71 
Barometer, 70, 87 
aneroid, 395 
mercurial, 376 

Barometric computation, example 
of, 381 

determination of heights, 382 
formula of Laplace, 379 
leveling, 374 

leveling, methods and accuracy 
of, 375 

notes and computation, 378 
tables of Guyot, 379, 383 
tables of Williamson, 379 
Bars, compensated base, 507 
Bars and steel tapes, relative merits 
of, for measurement of bases, 
499 

Base apparatus, contact-slide, 508 
Eimbeck duplex, 511 
iced-bar, 511 
Repsold, 514 

Base-bars, compensated, 507 
duplex, 511 

corrected length of, 523 
Base-line, 62 





INDEX. 887 


Base-line, cost, speed, and accuracy 
of measurement of, 516 
expansion of, 554 
laying out a. 505 
measuring a, 507 
Base map, 1 

Base-measuring apparatus, kinds 
of, 4gS 

Base measurement, 497 

measurement, accuracy of, 497,498 
measurement, correction of, for 
inclination, 519 

measurement, correction of, for 
sag, 521 

measurement, miscellaneous cor¬ 
rections to, 526 

measurement, probable error of, 
500, 501 

measurement, record of, 518 
measurement, reduction of, 517 
measurement, relative merits of 
bars and steel tapes for, 499 
measurement, sources of error in, 498 
measurement with steel tapes, 500 
Base, reduction of, to sea-level, 522 
reduction of broken, to straight line, 

5“ 6 

Base, selection of site for, 497 

summary of measures of sections of, 
5 23 

transfer of ends of, to triangulation 
signals, 524 
Basin, 131 
Basket, pack, 847 
Batson sketching-case, 166a 
Bay, 127 

Beam compass, 478 
Bechler, Wm. H., 283 
Bell odometer, 231 
Bench marks, 57, 316, 321 
Berliner Jahrbuch, 741, 751 
Bessel’s formulas, factors for comput¬ 
ing probable error by, 608 
solution of three-point problem, 188 
^-ino: ular precise level, 326 
Bites of serpents and insects, 860 
Black-prints of maps, 883 
Blankets, 850 
Blue-prints of maps, 883 
Board of Ordnance and Fortifica¬ 
tions, report of, 490 
Boards, plane-table, 39, 153 
Bohn, Dr. C., 490 
Boiling of atmosphere, 348 
Boiling-point of water, altitude by, 
402 

Boston level-rod, 313 


Boundary lines, conventional signs 
for, 469 

Bowhill, Maj. J. H., 490 
Brainard, F. R., 809 
Branner, John C., no, 490 
Bridges-Lee photo-theodolite, 297 
Brough, Bennett H., 490 
Buff and Berger precise level, 328 
Bumstead. A. H., 168 
Burns, 860 

Burt solar attachment, 781 
Butte, 129 

Cable, English equivalents of, 674 
Cadastral city survey, 64 
map, 106 

map of Baltimore, 66, 107 
maps, scales for, 447 
survey, 63 
surveys, 2, 103 

surveys, scale and cost of, 107 
Cadastre, 104 
Cadrer, 104 

Camera, photo-surveying, 298 

and its adjustments for photo¬ 
graphic longitudes, 794 
plates, projection of, 301 
Cameras, photographic, 865 
Camp cooking-stove, 831 
equipment, 831 
fire for cooking, 830 
fires, how to build, 830 
ground, an attractive, 815 
ground, selecting and preparing, 

814 

mess-boxes, 831 
mess kit, 831 

stoves, cots, and tables, 827 
subsistence in, 812 
tools, 833 

transportation, wagons for, 836 
Camping, underwear for, 851 
Canal line, topographic survey of, 55 
location, detailed contour survey 
for. 54 

location, topographic survey for, 
52 

preliminary map of, 56 
Canals, location of, 51 
Canteens, 854 
Canvas tent-floor. 826 
Canyon, 115, 119, 129 
Care of health, hints on, 853 
Carpenter, Wm. De Yeaux, 490 
Cartography, 404 

Casting duplicates of model maps, 

485 




888 


INDEX . 


Casts of model maps,materials for, 
486 

Cavalry sketch-board, 164, 284 
sketch-board, sketch map with, 285 
Celestial body, apparent displace¬ 
ment of, 688 

latitude and longitude, 679 
Celluloid for plane-table sheets, 175 
Center, reduction to, 608 
Centering, errors of, 579 
Centimeter, 677 

Central projection, Lambert’s sur¬ 
face-true, 410 
projections, 406 
Chain, 34, 100, 103 

Gunter’s, value of, in meters, 677 
measurements, 53 
traverse, 22 
in traverse, 196 
value of Gunter’s, 674 
Chaining, 234 

Chains, engineers’ and Gunter’s, 234 
Change, magnetic, 222 
Chart, isogonic, 222 
Chauvenet, William, 809 
Chauvenet’s Astronomy, 614, 617, 
678, 733, 747 

Check azimuths, correction for, in 
primary traverse, 539 
Chigres, 860 
Cholera-ban Is, 852 
Chronograph, longitude by, 748 
Chronometer, break-circuit, 749 
error, 752 
Chronometers, 76 
longitude by, 745 
Cinches for pack-saddle, 840 
Cinder-cone, 123 

Circuits of levels, adjustment of 
group of, 345 

Circumpolar stars, reduction of ob¬ 
servations on, 737 
Cistern of mercurial barometer, 376 
City surveys, 62 
Civil day, 683 
time, 683 

time, relations of, to sidereal and 
solar, 695 

Clamp-screws to fasten plane-table 
paper, 177 

Clarke, Col. A. R., 809 
Cliff, 115, 129 

Clinometer hand-level, Abney, 357 
Clothing, 850 

Coast and Geodetic Survey, United 
States, 63, 67, 222, 280, 343, 418, 
493- 810 


Coast coefficient of refraction by, 372 
and Geodetic Survey, cost, accur¬ 
acy, etc., of astronomic posi¬ 
tions of, 744 

cost, speed, and accuracy of base 
measurement of, 516 
cost and accuracy of triangulation 
of, 592 

duplex base-bars, 511 
level-rods of, 333 
leveling of, 325 

limit of precision in levelingin, 345 
method of leveling by, 332 
plane-table, 153 

solution of three-point problem, 
190 

speed and cost of levels, 351 
standards of measure, 517 
star tables from, 733 
tape sketchers of, 502 
telescopic alidades of, 157 
theodolites of, 555 
Coefficient of refraction, 371 
Colby, B. H., 490 
Colic, 857 

Collimation, error of, 581, 753 
Comparison of time, 774 
Compass, 103 
beam, 478 

-needle in plane-table traverse, 197 
prismatic, 214 

prismatic, on primary triangula¬ 
tion reconnaissance, 547 
prismatic, for traverse, 196 
Compensated base-bars, 507 
Computation of apparent declina¬ 
tion of star, 742 
barometric, 378 
of distances, 637 
example of barometric, 381 
of geodetic azimuths, 642 
of geodetic coordinates, 636 
of geodetic latitudes, 643 
of geodetic longitudes, 643 
by latitudes and departures, 213 
of latitudes and longitudes on 
primary traverse, 540 
of longitude, 757 
of photographic longitudes, 801 
of primary traverse, 538 
of precise levels, 333 
of trigonometric leveling, 368 
vertical angulation, 363 
Computations, factors for geodetic, 649 
Computing geodetic coordinates, 
formulas for, 638 
1 Comstock, Lt.-Col. C. B., 491 





INDEX. 


889 


Condition equations, 604, 612 
Conditions, general, applied to tri¬ 
angulation adjustment, 616 
local, applied to triangulation ad¬ 
justment, 616 
Cones, phase in tin, 560 
Conical projection, equal-spaced, 
415 

projection, equivalent, 415 
projection, intersecting, 414 
projection, Mercator’s, 414 
projection, tangent, 414 
Conical projections, 411, 412 
Connellsville coke map, 67 
Constants depending on spheroidal 
figure of earth, 672 
determination of level and mi¬ 
crometer, 732 
numerical, 672 
of transits, 752 
Constipation, 857 
Construction of contours, 460 
of hachure map, 461 
of maps, methods of, 449 
Constructive agencies, ill 
Contact-slide base apparatus, 508 
Contour construction, 451, 460 
grade, 51, 53 
interval, 63, 450 
lines, 53, 455 
map, 48 

map, shaded, 450, 453 
model maps, 487 
projection, 461 
sketch, 459 
sketching, 18 

survey for canal location, detailed, 

54 

survey of reservoir site, 59 
Contoured hill, 465 
Contours, 12, 450 
interpolating, 30 
sketch, 452, 455 
sketching, 37 

sketching, with aneroid, 399 
Control. 20 
horizontal, 10 
instrumental, 72 
land survey, 38 

by magnetic traverse with plane- 
table, 199 
plane-table, 23 
from public land lines, 36 
secondary, 34 
sheet, 37 

sheet, triangulation, 24 
topographic, 15 


Control, traverse, 13 

traverse for primary, 527 
trigonometric, 18 
vertical, 10 

Conventional signs, 463, 467, 469, 
471 . 473 . 475 

Convergence of meridians, 641 
of meridians, allowance for, in 
primary traverse, 539 
of meridians in map-projection, 405 
Conversion of English and metric 
measures, factors and loga¬ 
rithms for, 676 
Cooking fire for camp, S30 

outfits for transportation on 
backs, 833 
stove for camp, 831 
Coordinates, formulas for comput¬ 
ing geodetic, 638 
geodetic, 436, 636 
geographic, 636 
platting by rectangular, 212 
polar, 636 

for projection of maps, 416 
Corners, section, 37 
Corrasion, 111, 113, 114 
Corrected length of base, 523 
Correction of instrumental errors, 
580 

of base measurement for inclina¬ 
tion, 519 

of base measurement for sag, 521 
of base measurement for tempera¬ 
ture, 518 

Corrections for observers’ errors, 
578 

to primary traverse, 538 
Correlates, 627 

formation of table of, 614 
substitution in table of, 615, 632 
weighted, 635 

Cost of astronomic positions, , 44 
of base measurement, 516 
of cadastral surveys, 107 
of geographic surveys, 74, 75 
of large-scale topographic survey, 
67 

of leveling. 349, 351 
of primary traverse, 536 
of primary triangulation, 592 
Cost and scale of survey of India, 
67, 75 

of topographic surveys, 40 
of transportation, 814 
Cots, camp, 826 

Counting tape-lengths on primary 
traverse, 535 



INDEX . 


890 


Crater, 123 

Crayon-shading, relief by, 450-455, 

457 

Creek, 115, 131 
Crevasse, 125 

Criterion. Pierce’s, 604-606 
Cross-hairs, errors due to parallax 
of, 582 

Cross-tree pack-saddle, 838 
Croton water-shed, 67 
Culmination of star or planet, 682 
Culminations of Polaris, times of, 
7i3 

Culture, 5 

color for representation of, 464 
• conventional signs for public and 
private, 467 

Curvature, radius of, 369 
Curvature and refraction, 371, 550 
and refraction, elevation of instru¬ 
ments to overcome, 550 
and refraction in spirit-leveling, 
347 

and refraction in vertical angula¬ 
tion, 363 

Cutts. Richard D., 549, 809 
Cuts, 861 

Cylinder projection, Mercator’s, 412 
projections, 410 

Dam site, survey of, 58, 61 
Davidson, George, 809 
Day, sidereal, solar, mean, apparent, 
etc., 682 
Declination, 682 
arc of solar, 782 
magnetic, 221 

Declinations of stars, determination 
of apparent, 741 

Definitions of astronomic terms, 679 
Deflection angles in plane-table trav¬ 
erse, 200 

Departures and latitudes, 210 

and latitudes, computation by, 213 
and latitudes, for plotting transit 
notes, 212 

and latitudes, signs of, 213 
Developing photographic negatives, 
875 

Deville, E., 296, 491, 872 
Diagram, stadia reduction, 265 
Diamond hitch, throwing the, 841 
Diaphragms for lenses, 868 
Diarrhea, 857 
Diastrophic forms, 121 
processes, in 
Diastrophism, 112 


Diaz, Augustin, 491 
Differences in height between appar¬ 
ent and true level, 549 
of time, determining, 749 
Differential refraction, correction 
for, 738 

Direction theodolites, 555, 585 
Directions, measuring angles by, 

584 

Disintegration, in 
Displacement of celestial body, ap¬ 
parent, 688 

Distance, errors of measurement of, 
in primary traverse, 529 
Distances, 76 

computation of, 637 

and elevations for stadia readings, 250 

estimating, 283 

on horizontal from inclined stadia 
measures, 249 

inclined stadia reduced to hori¬ 
zontal by diagram, 264, 266 
with linen tape, 228 
measuring with gradienter, 272 
methods of measuring, 224 
reciprocal zenith at two stations, 
360 

by time, 226 

Distortion of lenses, 867 
Ditching tent, 825 

Diurnal aberration, correction for, 

754 

magnetic variation, 222 
Divergence of duplicate level-lines, 
330, 343 

Dividers, proportional, 478 
three-legged, 478 

Division of latitude level, value of, 

735 

Dome, 115 

Doolittle’s Practical Astronomy, 
678, 747 

Double-mounted plane-table paper, 

174 

Double leveling, sequence in, 329 
Double rodded levels, 322 
Double target level-rods, 333 
Double targeted levels, 319 
Doubtful observations, rejection of, 
604 

Douglas, E. M.. 231, 809 
on triangulation adjustment, 617 
odometer, 230 
Drafting instruments, 477 
Drawing, topographic, 449 
Drinking, excess in, 855 
water, 855 




INDEX. 


891 


Drowning, 858 
Dry plates, 869 
Dulles, Dr. Charles W., 809 
Dune, sand, 127 
Dunnington, A. F., 790 
Duplex base-apparatus of Eimbeck, 
512 

bars, 511 

Duplicate leveling with double rod, 
331 

leveling with single rod, 330 
Dysentery, 857 

Eating, excess in, 855 
Earth’s surface, areas of quadri¬ 
laterals of, 436 
Ecliptic, plane of, 68r 
Eimbeck, Wm., 809 
Eimbeck duplex base-apparatus, 512 
Elongation, azimuth at, 721 
Elongations of Polaris, times of, 713 
Elevation, differences of from, stadia 
measures, 260 

of instrument to overcome curva¬ 
ture and refraction, 550 
Elevations determined by stadia, 258 
and distances from stadia read¬ 
ings, 250 

from stadia readings, 249 
Elimination, solution by, 615 
Engineering news, 264 
spirit-leveling, 308 
surveys, 2 

Engineer’s chain, 234 
transit, 203 

transit, azimuth with, 708 
U. S. Army, level-rods of, 333 
U. S. Army, leveling of, 325 
U. S. Army, limit of precision of 
leveling of, 345 

U. S. Army, methods of leveling 
by, 332 

U. S Army, speed and cost of 
levels of, 351 

English equivalents of meter, 675 
linear measures, interconversion 
of, 673 

measures, metric equivalents of, 

^75 

and metric measures, interconver¬ 
sion of, 675 

and metric measures, factors and 
logarithms for conversion of, 
676 

square measures, interconversion 
of, 674 

Enthoffer, S., 491 


Ephemeris, Amerioan, 741, 747, 789 
Equation, correction for personal, 
774 

personal, 729 
Equations, angle, 619 
of condition, 604, 612 
formation of normal, 615 
side, 623 

solution of side and angle, 625 
Equator, plane of, 681 
Equatorial parallax, 6S9 
projections, 408 
projections, central. 408 
projections, orthographic, 408 
projections, stereographic, 409 
Equidistant flat maps, 411 
Equinox, autumnal, 681 
vernal, 681 
Equinoxes, 681 
Equipment for camp, 831 
Equivalents of English measures, 
673 

in metric system, 675 
of meter, English, 675 
scale, 446 
Erosion, 113 
aerial, 111 
aqueous, ill 

in hard and soft rocks, 120 
Erosive action, 113, 114 
Error of chronometer, 752 
of collimation, 581, 753 
probable of primary traverse, 537 
of run of micrometer, 583 
station, 775 

triangle of, in plane-table resec¬ 
tion, 190 

Errors, accidental, 580 
of aneroid, 395 
of centering, 579 
correction of instrumental, 580 
external, 729 
of graduation, 581 
due to inequality of pivots, 582 
instrumental, 729 
leveling, sources of, 339 
of latitude determination, 729 
of observation, 578 
kinds of, 602 
of observers, 729 
and their correction, 578 
due to parallax of cross-hairs, 582 
periodic, 580 
personal, 729 

in photographic longitudes, 
sources of, 804 
in primary traverse, 528 




892 


INDEX . 


Errors, systematic, 581 
due to twist in scaffolds, 579 
in vertical triangulation, 370 
Excess, spherical, 619, 638 
Expansion of base line, 554 
Exploratory and geographic sur¬ 
veys compared, 77 
map, 78, 79, 88, 89, 90 
map making, 23 
maps, scales for, 447 
plane-table and theodolite, 163 
surveys, 3, 76 

surveys, methods and examples 
of, 82 

surveys, use of sextant in, 777 
Exposures, photographic, 872 
External errors, 729 
projections, 407 
Extra-instrumental errors, 578 

Factors for conversion of English 
and metric measures, 676 
for geodetic computations, 649 
for reduction of transit observa¬ 
tions, 758 
Fall, 115 
Farrier’s kit, 848 

Fathom, English equivalents of, 674 
metric value of, 677 
Fevers, malarial, 856 
Field-work of geodetic astronomer, 
679 

instructions for primary triangu¬ 
lation, 590 

instructions as to topographic, 12 
of observing latitude, 730 
organization of, 22 
of photographic longitudes, 793 
Figure-adjustment, 616 
notation used in, 618 
routine of, 617 

Figure of earth, constants depend¬ 
ing on spheroidal, 672 
Figures for primary triangulation, 
548 

Films, celluloid and cut, 869 
cut celluloid, 871 
roll, 871 

Finding the stars, 686 
Fire for cooking in camp, 830 
wood, 830 

Fires, how to build camp, 830 

Fixing negatives, 878 

Flagmen on primary traverse, 532 

Flat, tidal, 127 

Flemer, J. A., 301, 491 

Flies, specifications for wall-tent, 821 


Flood-plain, 125 
Flooring, tent, 825 
Fly to wall-tent, 818 
Focusing-cloth, 866 
Food for tropics, 854 
Foot, English equivalents of, 673 
metric equivalents of, 675 
metric value of, 677 
Forage for animals, 849 
Formula for stadia with inclined 
sight, 246 

with perpendicular sight, 243 
Formulas for computing geodetic 
coordinates, 638 
fundamental astronomic, 684 
for trigonometric functions, 594 
Formulas for solution of right- 
angled triangles, 594 
France, survey of, 75 
Fremont, Captain J. C., 78 
Freshfield, Douglas W., 809 
Frome, Lieut.-Gen., 491 
Frostbite, 857 

Frost, effect of, on leveling, 341 
Fruit as food in tropics, 855 
Functions, fundamental formulas 
for trigonometric, 594 
logarithms of trigonometric, 217 
natural trigonometric, 275 
trigonometric, 594 
Furlong, English equivalents of, 673 

Gannett, Henry, 111, 491, 809 
traverse plane-table, 160 
Manual of Topographic Methods, 
730 

Gannett, S. S-, computation of base 
measurement, 520, 524 
Gauss’ logarithmic tables, 214 
Gelbcke, F. A., 401, 491 
Geodesic leveling, 307, 326 
Geodesy, 495 

works of reference on, 809 
Geodetic astronomer, field-work of, 
679 

astronomy, 679 
Astronomy, Hayford’s, 678 
azimuth, computation of, 642 
computations, factors for, 649 
coordinates, 436, 636 
coordinates, formulas for comput¬ 
ing, 638 

latitudes, computation of, 643 
latitude and longitude, 679 
longitudes, computation of, 643 
position, 23, 776 
survey, 1, 68 





INDEX. 


893 


Geographic coordinates, 636 

and exploratory surveys com¬ 
pared, 77 
map, 80, 81 
map-making, 23 
mapping by traversing, 196 
maps, 70 

maps, features shown on, 72 
maps, scales for, 447 
reports, 73 

survey, instrumental method, 69 
surveys, 1, 4, 68 

surveys, scale and cost of, 74, 75 
Geologic structure, 73 
Geological survey, United States, 
12, 18, 40, 71, 75, 82, 87, 105, 
231, 242, 343 
using aneroid on, 396 
barometric leveling of, 375 
bench marks of, 317 
chronometers of, 749 
color-scheme of maps bf, 464 
cost, accuracy, etc., of astronomic 
positions of, 745 

cost, speed, and accuracy of base 
measurement, 516 
cost and accuracy of triangulation 
of, 592 

method of topographic surveying, 
20 

instructions for leveling on, 320 
level-rods of, 333 
leveling of, 325 

limit of precision in leveling of, 
345 

long distance leveling of, 353 
number of sets of measures of an¬ 
gles of, 588 

plane-table used by, 152 
precise level of, 328 
primary traverse of, 536 
ration list of, 834 
reduction to center by, 608 
screw-clamps for fastening plane- 
table paper, 177 
speed and cost of levels of, 351 
standard scales of, 167 
tape stretchers of, 503 
telescopic alidades of, 158 
theodolites of, 556 
topographers of, 811 
traverse plane-table, 160 
zenith telescope of, 726 
Geology, relations of, to topog¬ 
raphy, 108 

Geometric locations, 14 
Gilbert, G. K., 380, 491 


Gillespie, Wm., 491 
Glacier, 123 

Glossary of topographic forms, 133 
Gnomonic projections, 406 
Gore, J. H., 809 

Government geographic surveys, 
scale and cost of, 74, 75 
surveys, 6 

value of topographic maps to, 6, 

7 . 8 

Gradation, 112, 113 
Grade contour, 51, 53 
Gradienter, leveling with, 372 

measuring distances with, 274 
Graduation, errors of, 581 
Graphic solution of three-point 
problem, 186 

Great Britain, Ordnance Survey of, 
67, 107 
survey of, 75 
Greenwich time, 747 
Gribble, Theo. Graham. 491 
Gunter’s chain, 234, 674 
value of, in meters, 677 
Gurley, W. & L. E., level-rods of, 
333 

solar, 781 

Guyot, A., 379, 491 

barometric tables, 379, 383 

Hachure construction, 451 
map, construction of, 461 
Hachured hill, 465 
Hachures, 12, 461 
shaded,463 
Hachuring, 462 
Hairs, stadia, 239 
Halation in plates, 870 
Hall, W. Carvel, 343, 353 
Hand, English equivalents of, 674 
Hand-level, 29, 355 
Hand recorder, 232 
Handicrafts, knowledge of, 812 
Hardy, A. S., 492 
Haupt, Lewis M., 491 
Hawkins, George T., record and 
reduction cf primary traverse, 
532 

Hayden map, 8; 
survey, 73, 75 

Hayes, Dr. C. Willard, 112, 122 
Hay ford, John F., 809 
Hayford’s Geodetic Astronomy, 
678. 733 

Health, hints on care of, 853 
Heat of sun, effect of on leveling, 
340 



894 INDEX. 


Heavenly bodies, apparent motion 
of, 680 
Hectare, 677 

Heights, barometric determination 
of, 382 

differences in, between apparent 
and true level, 549 
Heliotrope, automatic, 573 
hand-mirror, 570 
steinheil, 572 
telescopic, 570 
Heliotropes, 560, 566 
dimensions of mirrors of, 569 
Hemorrhage, 861 
Hergesheimer, E., 491 
Hilgard, J. E., 491, 810 
Hills, Capt. E. H., 492, 747, 794, 810 
Hodgkins, W. C., 492 
Homolographic projection, Moll- 
weide’s and Babinet’s, 412 
Horizon, 681 
Horizontal control, 10 
parallax, 688 
projections, central, 408 
projections, orthographic, 408 
projections, stereographic, 409 
Horn-protractor, 210 
Hour-angle, 681 

relations of, to time and right as¬ 
cension, 695 
Hour-circle, 681 
of solar, 782 
Howell, E. E., 483 
Hydrography, 5 

color for representation of, 464 
conventional signs for, 471 
Hypsography, 5 
color for representation of, 464 
conventional signs for, 473 
Hypsometric survey, 68 
Hypsometry, 305 

Iced-bar base-apparatus, 511 
Inch, English equivalents of, 673 
metric equivalents of, 675 
Inches, length of meter in, 674 
Inclination, correction of base meas¬ 
urement for, 519 

corrections for, in primary trav¬ 
erse, 538 

of plane-table board, 181 
Independent measures of angles, 
586 

India, cost and scale of survey of, 
f>7, 75 

Inequality of pivots, errors due to, 
582 


Information surveys, 3, 4 
Insect-bites, 860 

Instructions for field-work of pri¬ 
mary triangulation, 590 
for leveling, 320; for precise leveling, 
for primary traverse, 533 [329 

to topographer, 8 

Instrument, elevation of, to over¬ 
come curvature and refraction, 
550 

errors in primary traverse, 529 
manipulation of precise leveling, 
235 

Instrumental control, 72 
Instrumental errors, 578, 729 
errors and their correction, 580 
errors in leveling, 339 
methods in geographic surveys, 69 
Instruments, drafting, 477 
used in primary traverse, 529 
for primary triangulation, 553 
required on primary triangulation 
reconnaissance, 547 
used in topographic surveying, 21 
for triangulation reconnaissance, 
547 

Intensifier for negatives, 879 
Interconversion of English linear 
measures, 673 

of English and metric measures, 
675 

of English square measures, 674 
Interpolating contours, 30 
Intersecting on radial sight-lines, 
149 

Intersection, 182 
locating by, 34 
location by, 182 
on plane-table, 148 
from traverse, 24, 202 
Interval, contour, 450 
Intervals of sidereal and solar time, 
relations of, 695 

Intervisability of triangulation sta¬ 
tions, 549 

Inverse problem, 646 
Irregular method of topographic 
surveying, 19 
surveying, 93 

Irrigation canal, location of, 53 
Isochromatic plates, 869 
Isogonic chart of United States, 
222 

lines, 222 

Jacob’s staff. 214 
Jacoby, Henry S., 492 



INDEX. 895 


Johnson, J. B., 214, 249, 346, 492 
Johnson, Willard D., 91, 156 
Johnson, W. W., 810 
Johnson on leveling, 326 
plane-table movement, 156 

Kern-level, 325 
Kilometer, 677 
Kippax, Hargreaves, 786 

Lagoon, 127 

Lake survey, U. S., base-apparatus, 
5i4 

coefficient of refraction by, 372 
Lambert’s, equivalent conical pro¬ 
jection, 415 

surface-true central projection, 
410 

Land lines, control from public, 36 
lines, sketching over public, 37 
Office Manual, U. S., 242 
survey United States, 37 
survey, plats, 37 
survey control, 38 
survey control, topographic map 
on, 41 

survey map, 106 

survey map, United States, 40, 
105, 107 

Landreth, W. B., 242 

Laplace’s barometric formula, 379 

Large areas, projections for maps 

of, 437 , 445 
scale maps, 30 
Lash-rope, 839 
Latitude, 76 

approximate solar, 724 
astronomic, 679 

and azimuth, determination of 
with solar attachment, 789 
celestial, 679 

determination, errors and pre¬ 
cision of, 729 

by differences of zenith distances, 
728 

field-work of observing, 730 
geodetic, 679 

methods of determining, 723 
observations by Talcott’s method, 
corrections to, 738 
from an observed altitude, 725 
reduction of observations for, 743 
Latitudes, computation of geodetic, 
643 

and departures, 210 
and departures, computation of, 
2 T 3 


Latitudes and departures, for plat¬ 
ting transit notes, 212 
and departures, signs of, 213 
factors for computing geodetic, 
649 

and longitudes, computation of 
on primary traverse, 540 
Least squares, method of, 602 
Lee, Thomas J., 492, 810 
Legendre, rule of, 602 
Length of base, corrected, 523 
Lenses, accessories of, 867 
for cameras, 867 
Lettering, 477 
samples of, 475 
Level, 48, 57 

adjustments of, 308 
binocular precise, 326 
bubble, sensitiveness of, 339 
circuits, adjustment of group of, 
345 

constants in latitude observations, 
732 

differences in height between ap¬ 
parent and true, 549 
error, corrections for, 752 
hand, 29 

lines, divergence of duplicate, 343 
note-books, 322 
precise, 332; precise spirit, 327 
precise speaking, 335 
prism, 326, 333a ' 
rods, double-target, 333 
rods, single-target, 335 
rods, speaking, 270, 313 
spirit, 37, 63; trier, 752 
value of divisions of latitude, 735 
value of divisions of striding, 752 
Leveling, 23, 52, 305 

accuracy of trigonometric, 361 
barometric, 374 

computation of trigonometric, 368 
computation of precise, 333 

cost of, 349-35 1 

duplicate with double rod, 331 

with single rod, 330 

effect of atmospheric conditions on,, 

337 . 340 

engineering spirit, 308 
errors of, due to refraction, 342 
geodesic, 326 
geodesic spirit, 307 
with gradienter, 372 
instructions for, 320 
instructions for precise, 329 
length of sight in, 337 
limit of precision in, 344 
long-distance precise, 352 



896 INDEX - 


Leveling methods and accuracy of 
barometric, 375 
precise, 307, 325 
rods, target, 311 
sequence in double-rodded, 329 
sources of error in, 339 
speed in, 349-351 
spirit, 34 

thermometric, 402 
trigonometric, 359 
Levelman, work of, 318 
Levels, 151 

divergence in duplicate lines of, 
330 

double-rodded, 322 
double-targeted, 319 
hand, 355 

limit of error allowable in, 321 
method of running single lines of, 
317 

methods of running, 332 
platting profiles of, 324 
plumbing, 321 
precise, 64 

of primary quality, 320 
single-rodded, 320 
spirit, 22, 24 
trigonometric, 70 
Licka, J. L., 492 
Linen tape, 36 

tape distances with, 228 
Linear measures, interconversion of 
English, 673 
Lining-in, 192 

Lippincott, J. B , 53, 55, 62, 492 

Lists of stars, 731 

Local attraction, 223, 775 

conditions applied to triangula¬ 
tion adjustment, 616 
Locating by intersection, 34 
Location of canals, 51 
detailed contour survey for canal, 
54 

by intersection, 182 
of irrigation-canal, 53 
paper, 49 
bv resection, 185 

topographic survey for canal, 
52 

Locations, distribution of, 16 
geometric, 14 
number of, 16 

from plane-table traverse, 26 
three-point, 186 

Locke hand-level, using the, 356 
Logarithms for conversion of Eng¬ 
lish and metrical measures, 676 


Logarithms of numbers, 215 
of trigonometric functions, 217 
Long-distance precise leveling, 352 
Longitude, 76 
astronomic, 679 
celestial, 679 
by chronograph, 748 
by chronometers, 745 
computation, 757 
determination of, 744 
geodetic, 679 
by lunar distances, 746 
Longitudes, computation of geo¬ 
detic, 643 

factors for computing geodetic, 
649 

field-work of photographic, 793 
and latitudes, computation of, on 
primary traverse, 540 
by photography, computation, 801 
by photography, measurement of 
negative, 797 

by photography, precision of, 
808 

by photography, sources of error 
in, 804 

Lunar distances, longitudes by, 
746 

photography, 76 

McCulloch, E., 264 
McMaster, Jno. B., 492 
Magnetic attraction, local, 223 
declination, 221 

diurnal, secular and annual vari¬ 
ation, 222 

needle in plane-table traverse, 
197 

Malarial fevers, 856 
Manual, Gannett’s, 730 
of U. S. Land Office, 242 
Map of Baltimore, 66, 107 
base, 1 

cadastral, 106 
of canal, preliminary, 56 
completed topographic, 31 
construction, methods of, 449 
construction, photograph used in, 
293 

construction of hachured, 461 
construction from photographs, 
303 

contour, 48 

correctness of topographic, 15 
exploratory, 78, 79, 88, 89, 90 
geographic, 80, 81 
Hayden, 81 




INDEX . 897 


Map modeling, 480 
mother, 1, 9 
mould of model, 485 
military reconnaissance, 97, 98 
siege, 101, 102 
topographic, 100, 101 
plane-table sketch, 71 
projecting the photographic, 300 
projection, 405 

public uses of topographic, 6 
reconnaissance sketch with range¬ 
finder, 285 

shaded contour, 450, 453 
skeleton model, 486 
topographic, 5, 39, 62, 109, 147 
U. S. land survey, 106 
on land survey control, 41 
Wheeler, 80 

Map-making, exploratory, 23 
geographic, 23 

Mapping, traversing for geographic, 
196 

Maps, accuracy desirable in topo¬ 
graphic, 8 

accuracy of topographic, 9, 15 
blue- and black-prints of, 883 
color-scheme of U. S. geological 
survey, 464 

construction of relief, 480 
coordinates for projection of, 416 
equidistant flat, 40 
features shown on geographic, 

7 2 

features shown on topographic, 5 
features shown on geographic, 70 
of large areas, projections for, 
437-445 

large scale, 30 

materials for modeling, 485 

methods of making model, 483 

military, 6 

model, 449 

contour, 487 

materials for casts of, 486 
model and relief, 478 
preliminary, 9 

projections of upon a po.yconic 
development, 418 
reconnaissance, 9 
relief, 449 

scale, cost and relief of geographic, 

75 

scale, cost and relief of detailed 
topographic, 67 

scales for exploratory, geographic, 
topographic, and cadastral, 447 
uses of model, 479 


Maps, value to government of topo¬ 
graphic, 6, 7, 8 

value to state of topographic, 6, 

7 , 8 

Mark, azimuth, 707 
Marks, permanent on primary tra¬ 
verse, 532, 535 
station and witness, 575 
surface, 575 
underground, 575 
Marsh, 127 

Mattress for camp, 850 
Mean day, 682 
time, 6S3 

Measure of base, reduction of, to sea 
level, 522 

Measured base, transfer of ends of 
to signals, 524 

Measuring angles by directions, 5S4 
by repetitions, 584 
apparatus for bases, kinds of, 498 
a base, 507 

distances with gradienter, 272 
distances with linen tape, 228 
methods of, 224 

horizontal angles, methods of, 584 
precautions in, 577 
Measurement of angles indepen¬ 
dently, 586 
by series, 586 
base, 497 

base, accuracy of, 497, 498 
base, correction of, for inclination, 
519 

base, correction of, for sag, 521 
base, miscellaneous corrections to, 
526 

base, probable error of, 500, 501 
base, record of, 518 
base, with steel tapes, 500 
base, sources of error in, 498 
of bases, cost, speed, and accuracy 
of, 516 

of distance, errors of, in primary 
traverse, 529 
by pacing and time, 226 
horizontal stadia, 244 
with odometer, 229 
reduction of base, 517 
reduction of, on slope with stadia, 
248 

with wheel, 229 
Measurements, chain, 53 
Measures of angle, number of sets, 
587 

English equivalents of metric, 

675 





898 INDEX 


Measures, factors and logarithms 
for conversion of English and 
metric, 676 

interconversion of English linear, 
673 

metric, 675 
square, 674 

metric equivalents of English, 675 
of sections of base, summary of, 
523 

with stadia, effects of refraction 
on, 266 

with stadia, horizontal distances 
from inclined, 249 
Medical hints, 856 
Medicine chest, 861 
Memoirs, military, 93, 102 
Mendell, G. H., 492 
Mendenhall, T. C., 499 
Mens backs, packing on, 845 
Mercator’s cylinder projection, 412 
Mercator’s conical projection, 414 
projection, 411 
Mercurial barometer, 376 
Meridian attachment, Smith, 785 
reduction to, for latitude, 739 
transits, time by, 703 
zenith distance, latitude by, 723 
Meridians, allowance for conver¬ 
gence of, in primary traverse, 539 
convergence of, 641 
convergence of, in map-projection, 
405 

Merriman, Mansfield, 810 
on least squares, 614, 617 
Mesa, 129 

Mess-boxes for camp, 831 
kit for camp, 831 
Meter, 677 

English equivalents of the, C75 
length of, in inches, 674 
Meters, conversion of, to statute 
and nautical miles, 676 
value of Gunter’s chain in, 677 
Method of least squares, 602 
Methods, astronomic, 678 
Metric equivalents of English meas¬ 
ures, 675 

measures, English equivalents of, 

675 

measures, factors and logarithms 
for conversion of English and, 

676 

measures, interconversion of Eng¬ 
lish and, 675 

Micrometer constants in latitude 
observation, 732 


Micrometer, error of run of, 583 
microscopes, 556 
theodolites, 554 
Microscopes, micrometer, 556 
Middleton, Reginald E., 241, 492 
Mile, English equivalents of, 673 
metric equivalents of, 675 
Military maps, 6 
memoirs, 93, 102 
reconnaissance map, 97, 98 
sketch, 99 

sketch with guide-map, 95 
sketch without guide-map, 95 
siege-map, 101, 102 
surveys, 92 

topographic map, 100, 101 
Millimeter, 677 
Mindeleff, Cosmos, 483, 492 
Mirror heliotropes, simple, £70 
Mirrors, dimensions of heliotrope, 
569 

Mississippi River Commission, U. S. 

standard of measure, 517 
Missouri River Commission, U. S. 

standard of measure, 517 
Model contour maps, 487 
duplicating by casting, 485 
maps, 449, 478 
materials for casts of, 486 
methods of making, 483 
mould of, 485 
uses of, 479 
maps, skeleton, 486 
Modeling the map, 480 
maps, materials for, 485 
Models, 479 

Mollweide’s homolographic projec¬ 
tion, 412 

Moon exposures for photographic 
longitude, 794 
Moore pack-saddle, 839 
Moraine, 123 

Morales y Ramirez, D. Jose Pilar, 492 
Morrison, G. James, 492 
Mother-map, 1 , 9 

Motion, apparent of heavenly bodies, 680 
Mountain, 115 
range, 131 
ridge, 117 
volcanic, 123 

Mountains, care of health in, 853 
Mould of model map, 485 
Movements, plane-table, 153 
Myer’s formula for time, 704 

Nadir, 681 

Natural logarithm functions, 275 




INDEX. 


899 


Natural sines and cosines, 275 
tangents and cotangents, 277 
Nautical miles, 676 
Neck, 140 
Needle-points, 179 
Needle, traversing with plane-table 
and magnetic, 197 
Negative varnish, 881 
Negatives, defects in, 880 
photographic, 879 
Neve, 141 

New York level-rod, 313 

Noe, de la and de Margerie E., 492 

Normal equations, 627 

equations, substitution in, 615, 632 
Notation, angle and side, 618 
astronomic, 683 
used in figure-adjustment, 618 
Notch, 141 
Note-books, 322 
Notes, barometric, 378 
level, 323 

platting transit, 211 
of primary traverse, 536 
of transit traverse, 208 
triangulation, 588 
vertical angulation, 362, 367 
Numbers, logarithms of, 215 
Numerical constants, 672 
Nunatak, 141 

Oblique plane triangles, solution 
of, 597 

Observation, errors of, 578 
kinds of errors of, 602 
Observations on circumpolar stars, 
reduction of, 737 
for latitude, reduction of, 743 
by Talcott’s method, corrections 
to, 738 

rejection of doubtful, 604 
for time, record of, 754 
reduction of, 752 
weighted, 633 

Observed quantity, probable value 
of, 603 

Observers’ errors, 729 

errors, and their correction, 578 
Observing scaffolds, 565 
for time, 751 
Ocean, 127 
Ockerson, J. A., 492 
Odometer, 39, 48, 93, 100, 229 
conversion table, 233 
traverse, 21, 22, 24 
in traversing, 196 
Ogden, Herbert G., 492 


Opaque signals, 560 
Open-country sketching, 28 
surveying, 23 

Optical illusions in sketching topog¬ 
raphy, 44 

Ordnance Survey of Great Britain, 
67, 107, 492 

Organization of field survey, 22 
Orientation, 181, 182 
Orienting traverse plane-table, 198 
Orthochromatic plates, 869 
Orthographic projections, 407 

Paced traverse, 21, 22, 28 
Pacing, 36, 70, 87, 93 

measuring distances by, 224 
in traverse, 196 
Pack animals and saddles, 837 
basket, 847 
cinches, 840 

mule, ability of, as climber, 813 
loading the, 843 
Packing on men’s backs, 845 
Packmen, 847 

Palm, English equivalents of, 674 
Panniers, 839 
Pantograph, 477 

Paper, fastening to plane-table 
board, 177 
location, 49 
plane-table, 174 
Parallax, 688 

of cross-hairs, errors due to, 582 
of sun, tables of, 691 
Parallel projections, 407 
Patterson, Lieut.-Col. Wm.,493 
Peak, 131 
Pedograph, 291a 
Pelleton, A., 493 

Pencil-holders, 179 [178 

Pencil manipulation of, in plane-tabling, 
sharpeners, 179 
Perkiomen water-shed, 67 
Periodic errors, 580 
Permanent marks on primary trav¬ 
erse, 535 

Personal equation, 729 

equation, correction for, 774 
errors, 729 

Perspective projections, 406 
Peters, Wm. J.. 87, 748, 794 
Phase in reflecting signals, 560 
in tin cones, 560 
Philadelphia level-rod, 313 
Photograph used in map-construc¬ 
tion, 293 

Photographic exposures, 872 




900 


INDEX . 


Photographic longitudes, camera 
and adjustments, 794 
longitudes, computation of, 801 
longitudes, field-work of, 793 
longitudes„measurement of nega¬ 
tive, 797 

longitudes, precision of, 808 
longitudes, sources of error, 804 
map, projecting the, 300 
negatives, fixing, 878 
prints, defects in. 883 
surveying cameras, 866 
Photographs, developing plates, 875 
fixing and toning, 880 
lunar, 76 

Photography, uses of, in surveying, 
864 

Photo-surveying, 292 
camera and plates, 298 
lenses for, 868 

and plane-table surveying com¬ 
pared, 292 

Photo-theodolite, 297 
Photo-topographv, principles of, 296 
Photo-topographic survey, field¬ 
work of. 299 

Physiographic processes, no 
processes, classification of, 112 
Physiography, no 
Pierce’s Criterion, 604-606 
Pierce, Josiah, Jr., 18r, 493 
Pike, Captain Zebulon M., 77 
Pirating streams, 117 
Pivots, correction for inequality of, 
752 

errors due to inequality of, 582 
Plain, flood, 125 
Plains, sketching, 35 
surveying, 33 
Plane angles, 63S 
of ecliptic, 6S1 
of equator, 681 
survey, 68 

and topographic surveying, 146 
triangles, solution of, 596 
Planes of reference, astronomic, 
681 

Plane-table, 48, 49, 52, 55, 62, 69, 100 
board, 39 

fastening paper to, 177 
inclination of, 181 
Coast survey, 153 
control, 23 

folding exploratory, 163 
Gannett traverse, 160 
intersection on, 148 
location with by intersection, 183 


Plane-table, manipulation of pencil 
and straight-edge on, 178 
movement, Johnson, 156 
movements, 153 
paper, 174 

and photo-surveying compared, 
292 

sheets, preparation of, 175 
range-finding with, 290 
setting up, 180 
sheet, 23 
sketch map, ~]i 
station on mountains, 851 
surveying, 147 
traverse by deflections, 200 
traverse, locations from, 26 
traverse with magnetic needle, 
197 

traverse, roads resulting from, 26 
triangulation, 12, 23, 24 
triangulation, diagram of, 25 
triangulation, reconnaissance and 
execution of, 149 
triangulation, secondary, 150 
triangulation from tertiary sketch 
points, 151 

tripods and boards, 153 
Plane-tables, varieties of, 152 
Planet, culmination of star or, 682 
transit of star or, 682 
Plateau, dissected, 129 
Plates, dry, 869 

various photographic, 869 
Plats, land, 37 

Platting paced and timed surveys, 
227 * * 

profiles, 324 

by rectangular coordinates, 212 
transit notes by latitudes and de¬ 
partures, 212 
with protractor, 210 
Poisoning, S57 
Polar axis of solar, 784 
coordinates, 636 
projections, central, 408 
projections, orthographic, 408 
projections, stereographic, 409 
Poles, specifications for shelter-tent, 
824 

for wall-tent, 822 
Polaris, 76 
azimuths of, 714, 716 
latitude by altitudes of, 723 
times of culminations and elonga¬ 
tions of, 713 

Polyconic development, projection 
of maps upon, 418 




INDEX. 


901 


Polyconic projection, 416 

projection, construction of, 416 
Position, geodetic, 23, 776 
Positions, astronomic, 76 
Powell, John W., 109, 493 
Powell survey, plane-table used on, 

152 . 

Precautions in measuring horizontal 
angles, 577 

Precise level, binocular, 326 
manipulation of, 335 
rods, 332, 336 

leve’ing,307,325; instructions for,329 
leveling, long-distance, 352 
levels, 64; computation of, 333 
sneaking level-rods, 336 
spirit-level, 327 

Precision of latitude determination, 
729 

limit of, in leveling, 344 
of photographic longitudes, 808 
Preliminary maps, 9 
map of canal, 56 
Pressure, atmospheric, 374 
Primary azimuths, 719 
control, traverse for, 527 
triangulation, 37, 545 
triangulation, accuracy and prob¬ 
able error of, 553-592 
friangulation, cost, speed, and 
accuracy of, 592 
triangulation, error of, 496 
triangulation figures, 548 
triangulation, instructions for 
field-work of, 590 
triangulation, instruments for, 553 
triangulation, reconnaissance for, 
546 

triangulation, theodolite for, 554 
traverse, 34, 37 

traverse, adjustment of, to astro¬ 
nomic positions, 542 
traverse, allowance for convey¬ 
ance of meridians in, 539 
traverse, azimuth on, 535 
traverse, computation of, 538 
traverse, computation of latitudes 
and longitudes on, 540 
traverse, correction for check 
azimuths, 539 

traverse, corrections to, 538 
traverse, cost, speed, and accuracy 
of, 536 

traverse, errors of, 497, 528 
traverse, instructions for, 533 
traverse, instruments used in, 
529 

traverse, method of running, 531 


Primary traverse, permanent marks 
on, 532, 535 

traverse, personnel of party on, 
53i, 535 

traverse, record of, 536 
traverse, record and reduction of, 
532 

traverse, tape-lengths on, 535 
Printing photographs, 880 
Prints of photographs, defects in, 883 
Prism level, 3:6, 333 a 
Prismatic compass, 70, 87, 214 
compass on primary triangulation 
reconnaissance, 547 
traverse, 196 

Probable error of arithmetic mean, 
607 

of base measurement, 516 
by Bessel’s formula, factors for 
computing, 608 

in primary triangulation, 553-592 
on primary traverse, 537 
Probable value of an observed quan¬ 
tity, 603 

Problem, inverse, 646 
three-point, 600 
Profile-platting, 324 
Projection, conical, 411, 412 
Babinet’s homolographic, 412 
Bonne’s, 416 

construction of polyconic, 416 
of contours, 461 
cylinder, 410 
equal-space conical, 415 
equivalent conical, 415 
intersecting cone, 414 
Lambert’s surface-true central, 410 
of maps, coordinates for, 416 
for maps of large areas, 437-445 
of maps upon a polyconic devel¬ 
opment, 418 
Mercator’s, 411 
conical, 414 
cylinder, 412 

Mollweide's homolographic, 412 
platting triangulation stations on, 
436 

polyconic, 416 
Sanson-Flamsteed, 412 
star, 410 

tables, method of using, 435 
tangent cone, 414 
Van der Grinten, 413 
Projections, map, 405 
circular, 412 
central, 406 
horizontal, 408 
polar, 408 






902 


INDEX. 


Projections, equatorial, 408 
external, 407 
gnomonic, 406 
orthographic, 407 
orthographic equatorial, 408 
orthographic horizontal, 408 
polar, 408 
parallel, 407 
perspective, 406 
stereographic, 407 
equatorial, 409 
horizontal, 409 
polar, 409 

Proportional dividers, 478 
Protractor for platting transit notes, 
210 

Protractors, 210 
Provisions, 833 
Prussia, survey of, 75 
Pyro-developing solution, 877 

Quadrilaterals of earth’s surface, 
areas of, 436 
Quadripod signals, 561 
Qualtrough, Lieut. E. F., 810 
Quartermasters’ Department, U. S. 
Army, 280 

Radial lines, intersecting on, 149 
sight-lines, 148 
Radius of curvature, 360 
Railway location, detailed topo¬ 
graphic survey for, 49 
location, topography for, 47 
reconnaissance, 47 
topography for, 48 
Range, mountain, 131 
Range-finder, 93, 100 
sketch map with, 285 
surveying with, 283 
for traverse, 284 
Weldon, 286, 288 
Range-finding, 282 

accuracy and difficulties of, 289 
with plane-table, 290 
Ranging-in, 192 
Ration, 833 
Ray-filter, 869 
Raymond, Wm. G., 493 
Reber, Lieut. Samuel, 810, 874 
Reconnaissance maps, 9 
military, 95 

for plane-table triangulation, 149 
for primary triangulation, 546 
railway, 47 

sketch map with range-finder, 285 
visual, 51 

Record of base measurement, 518 
of primary traverse, 532, 536 


Recor of time observations, 754 
of triangulation observations, 588 
Recorder, hand, 232 

on primary traverse, 531 
Rectangular coordinates, platting 
by, 212 

Reduction of base to sea level, 522 
of broken base to straight, 526 
to center, 608 

of latitude observations, 743 
of primary traverse, 532 
Reference planes of astronomy, 681 
works on geodesy and astronomy 
809 

works on topography, 490 
Regular method of topographic sur¬ 
veying, 19 

Reflecting signals, 560 
signals, phase in, 560 
Refraction, 689 
coefficient of, 371 
correction for differential, 738 
and curvature, 371, 550 
table of, 549 

elevation of instruments to over¬ 
come, 550 

in spirit-leveling, 347 
in vertical angulation, 363 
effects of, on leveling, 342 
effects of, on stadia measures, 266 
Relief, 12 

color for representation of, 464 
conventional signs for, 473 
by crayon-shading, 450-455, 457 
of geographic maps, 75 
maps, 449, 478 
maps, construction of, 480 
representation of, 450 
Repairs to camp-conveyances, 848 
Repeating theodolites, 586 
Repetitions of angles, number of, 

585 

measuring angles by, 584 
Reports, geographic, 73 
Representation of culture, color for, 
464 

of hydrodrophy or water, color 
for, 464 

of hypsography or relief, color 
for, 464 

of woods, color for, 464 
Repsold base-apparatus, 514 
Resecting to obtain position, 150 
Resection, 182, 185 
location by 185 
triangle of error in, 190 
Reservoir site, contour survey of, 59 
survey, 60 





INDEX. 


9°3 


Reservoirs, surveys for, 57 
Residuals, 604 

Respiration, restoring artificially, 
859 

Richards, Col. W. H., 493 
Ridge, alluvial, 125 
Right-angled triangles, formulas for 
solution of, 594 
Right ascension, 681 

relations of, to time and hour 
angles, 695 
River, 115, 125 

Roads resulting from plane-table 
traverse, 26 

Rod, English equivalents of, 673 
Rod-errors in leveling, 342 
Rodman, work of, 317 
Rods, double-target level, 333 
precise level, 332 
precise speaking level, 336 
single-taiget level, 335 
speaking level, 313 
stadia, 239 
stadia and level, 270 
target leveling, 311 
Rood, English equivalents of, 674 
Rope, lash, 839 

Route survey, exploratory, 88, 89 
Rule, slide, 168 

Run of micrometer, error of, 583 
Runge, C., 810 

Running levels, methods of, 332 

Saddler’s kit, 848 
Saddles, pack, 837 
Saegmuller & Co., base-bars, 508 
Safford, T. H., 810 
Safford’s Catalogue of Stars, 741 
Sag, correction of base measure¬ 
ment for, 521 
Sand-dune, 127 

Sanson-Flamsteed projection, 412 
Scaffolds, errors due to twist in, 579 
observing, 565 
Scalds, 860 

Scale of cadastral surveys, 107 
of detailed topographic maps, 67 
equivalents, 446 
of geographic surveys, 74, 75 
for platting transit notes, 210 
Scales, 166c 

for exploratory, geographic, topo¬ 
graphic, and cadastral maps, 447 
special, 166 d 
steel, 478 
Scarp, 131 

Schott, Chas. A., 493, 810 


Schweizerische Landesvermessung, 
493 

Screw-tacks to fasten plane-table 
paper, 177 

Sea level, reduction of base to, 522 
level, reduction of primary trav¬ 
erse to, 538 

Secondary azimuths, 712 
control, 34 

plane-table triangulation, 150 
traverse, 35 
Section corners, 37 
Secular magnetic variation, 222 
Self-reading level and stadia rods, 269 
Series, measurement of angles by, 586 
Serpent bites, 860 

Sets of measures of angles, number 
of, 587 

Setting up plane-table, 180 
Sextant, 87, 777 
adjustment of, 778 
parts of, 778 
using the, 780 
Shade-line construction, 451 
Shaded contour map, 450, 453 
hachures, 463 

Shading, relief by crayon, 450, 455, 
457 

Sheet, adjusted sketch, 27 
control, 37 
plane-table, 23 
sketch, 24 
traverse, 24 

triangulation control, 24 
Sheets, sketch, 35 

Shelter-tent poles, specifications for, 
824 

tents, specifications for, 822 
Shutters for lenses, 869 
Sibley tent-stoves, specifications for, 
829 

Side and angle equations, solution 
of, 625 

Side equations, 623 
Sidereal day, 682 
time, 683 

time, relations of, to civil and 
solar, 695 

time, relations of, to right ascen¬ 
sion and hour angle, 695 
Siege-map, military, 101, 102 
Sight alidade, 36 

alidade, in plane-table traverse, 197 
alidade, vertical angle, 202 
alidades, 161 

length of, in leveling. 337 
lines, drawing radial, 148 





9°4 


INDEX. 


Sign, conventional, 467, 469, 471, 
473 , 475 

Signals, night, 574 
opaque, 560 
phase in reflecting, 560 
reflecting, 560 

transfer of ends of base to, 524 
triangulation, 559 
tripod and quadripod, 561 
Signs, conventional, 463 
Single-target level-rods, 335 
Site for base, 497 
Skeleton model maps, 486 
Sketch-board, 69, 93 
cavalry, 164, 284 
Sketch contour, 459 
contours, 452, 455 
map, 71 

military reconnaissance, 99, 102 
points, plane-table triangulation 
from tertiary, 151 
sheet, 24; adjusted, 27 
sheets, 35 
Sketching, 39, 63 

art of topographic, 40 
board, cavalry, 164 
case, Batson, 166a 
contours, 18, 37 
with aneroid, 399 
open country, 28; plains, 35 
over public land lines, 37 
quality of, 16 
topographic, 13, 15 
vertical angulation in, 363 
woodland, 35 
Sleeping-bags, 850 
Slide-rule, 168 
uses of, 169 
Slope-board, 55 
Slough. 127 
Smith, Glenn S., 792 
Leonard S., 266 493 
meridian attachment, 785 
Lieut. R. S., 493 

Smithsonian Geographical Tables, 
2 75> 4i8, 596, 702 
Snake bites, 860 
Sod-cloth for tent, 826 
Solar, adjustment of Burt, 782 
alidade, Baldwin, 792 a 
traversing with, 792 
attachment, 781; to alidate, 791 
Smith meridian, 785 
azimuth, approximate, 708 
day, 682; time, 683 
latitude,, approximate, 724 [695 

time, relations of, to sidereal and civil, 


Solution by elimination, 615 
Solutions, developing, 876 
Span, value of the, 674 
Speaking level-rods, 313 
level-rods, precise, 336 
stadia and level-rods, 269 
Specht, George G., 492, 493 
Special surveys, 7 

Specifications for Sibley tent-stoves» 
829 

for shelter-tent poles, 824 
for shelter-unts, 822 
for wall-tent poles, 822 
for wall-tents and flies, 820 
Speed of astronomic determinations, 
744 

of base measurement, 516 
in leveling, 349, 351 
of primary traverse, 536 
of primary triangulation, 592 
Spherical angles, 638 
angles, corrected, 633 
excess. 619, 638 

Spheroidal figure of earth, constants 
depending on, 672 
Spirit-level, 37, 63 
lines, 20 
precise, 327 
Spirit-leveling, 34, 306 
engineering, 308 
geodesic, 307 
precise, 307 
Spirit-levels, 22, 24 
Spit, 127 
Sprains, 861 

Square measures, interconversion of 
English, 674 

Stadia, 34, 55, 58, 63, 87, 93 

distances on incline reduced to 
horizontal by diagram, 264, 266 
elevations from, 258 
formula with inclined sight, 246 
formula with perpendicular sight, 
243 

hairs, 239 

measurement on horizontal, 244 
measurement on slope, 248 
measures, 21 

measures, determining horizontal 
distances from inclined, 249 
measures, differences of elevation 
from, 260 

measures, effects of refraction 
on, 266 

readings, distances, and eleva¬ 
tions from, 250 

readings, elevations from, 249 




INDEX . 


9 ° 5 


Stadia, reduction diagram, 259, 264, 
265, 266 
rods, 239, 269 
rods, speaking, 270 
surveying, 238 
tachymetry with, 238 
tachymetry, accuracy and speed 
of, 240 

traverse, 22, 24, 237 
in traverse, 196 
topography with, 237 
Standard, reduction of base to, 517 
time, 683 

Star factors. A, B, C, 758 

or planet, culmination of, 682 
projection, 410 
transit of, 682 

Stars, determination of apparent de¬ 
clinations of, 741 
finding the, 686 
lists of, 731 

Safford’s Catalogue of, 741 
State, value of topographic maps 
to, 6, 7, 8 

Station adjustment, 610 

adjustment, routine of, 610 
error, 775 
marks, 575 

Stations, astronomic, 34 

intervisability of, for triangula¬ 
tion, 549 

triangulation, platting on projec¬ 
tion, 436 

Statute miles, conversion to meters, 
676 

Steel scales, 478 
Steel tapes, 501 

tapes and bars, relative merits of, 
for measurement of bases, 498 
tapes, base measurement with, 500 
tapes, for measuring in primary 
traverse. 531 
Steinheil heliotrope, 572 
Stereographic projections, 407 
St. Louis, survey of, 104, 107 
Stops to lenses, 868 
Stove, cooking for camp, 831 
Stoves, specifications for Sibley 
tent, 829 
camp, 826 

Straight-edge, manipulation of, in 
plane-tabling, 178 
Stretchers, tape, 501 
Striding-level on alidade, 159 
Subsistence of survey party in field, 
813 

Suffocation, 858 


Sun. parallax of, 689 

tables for parallax of, 691 
Surface of earth, areas of quadri¬ 
laterals on, 436 

forms, representation of, by relief, 
450 

Surgery, veterinary, 849 
Surgical advice, 860 
Survey of Baltimore, 64, 66, 67, 104, 
107 

Survey, cadastral, 63, 103 
cadastral of city, 64 
of canal line, topographic, 55 
city, 62 

cost of large-scale topographic, 
67 

of dam site, 58, 61 
elements of a topographic, 14 
exploratory route, 88, 89 
field-work of photo-topographic, 
299 

geodetic, 68 
Hayden, 73, 75 
hypsometric, 68 
object of topographic, 5 
organization of field, 22 
plane, 68 
reservoir, 60 

of reservoir site, contour, 59 
of St. Louis, 104, 107 
topographic of city, 64 
town site, 62 
trigonometric, 20 
U. S. land, 40 
of Washington, 63 
Wheeler, 73, 75 

Surveying, economy in topographic, 
22 

Geological Survey method of topo¬ 
graphic, 20 

instruments, automatic, 291a 
instruments used in topographic, 21 
irregular, 93 

irregular method of topographic, 

19 

methods of topographic, 18 
open-country, 23 
photographic, 292 
ph otographic, cameras for, 866 
uses of photography in, 864 
plains, 33 

plane and topographic, 146 
plane-table, 147 

with plane-table and camera com¬ 
pared, 292 

with range-finder, 283 

regular method of topographic, 19 



go 6 


INDEX . 


Surveying, speed in topographic, 22 
with stadia, 238 
woodland, 33 

Surveyor’s chain, value of, 674 
Surveys, cadastral, 2 
classes of, I 

comparison of exploratory and 
geographic, 77 
cost of topographic, 40 
engineering, 2 
exploratory, 3, 76 
geodetic. 1 
geographic, 1, 4, 68 
information, 2, 3 

instrumental methods in geo¬ 
graphic, 69 

methods and examples of explo¬ 
ratory, 82 
military, 92 

plane-tables used on Geological 
and Powell, 152 
prosecution of topographic, 5 
for reservoirs, 57 
scale and cost of cadastral, 107 
scale and cost of geographic, 74, 75 
special, 7 

use of sextant in exploratory, 777 
topographic, I, 3, 4 
traverse, 195 

United States public land, 107 
Swamp, 125, 127 

Symbols, miscellaneous, conven¬ 
tional, 469 

Systematic errors, 581 

Table of correlates, formation of, 
614 

of correlates, substitution in, 615 
traverse, 36 
Tables, camp, 826 
Tachygraphometer, 280 
Tachymeter, Wagner-Fennel, 280 
Tachymetry, 236 

accuracy and speed of with stadia, 
240 

angular, 272 
with stadia, 238 
Tachymetry, 236 

Talcott’s method, corrections to ob¬ 
servations for latitude by, 738 
Tangent-cone projection, 414 
Tape, 34 

distances with linen, 228 
lengths, corrections to, in primary 
traverse, 538 

counting on primary traverse, 535 
linen, 36 


Tape-stretcher, simple, 504 
for use on railroads, 504 
stretchers, 501 

Tapemen on primary traverse, 532 
Tapes, base measurement with steel* 
500 

for measuring in primary traverse, 
531 

steel, 501 

and bars, relative merits of, for 
measurement of bases, 499 
Target leveling-rods, 311 
stadia and level rods, 269 
Teeple, Jared, 493 
Telephoto combination lenses, 869 
Telescope, astronomic transit and 
zenith, 726 
zenith, 723 

Telescopic alidade, 21 

alidade, adjustments of, 159 
alidade in plane-table traverse, 
200 

alidades, 157 
heliotropes, 570 

Temperature, correction of, to base 
measurement, 518 
correction for, in primary traverse, 
533 

Tent-ditching and -flooring, 825 
Tent, erecting the, 825 
sod-cloth, 825 

stoves, specifications for Sibley, 
829 

Tents, 817 
A, 819 

specifications for flies and wall, 
S20 

Tents, specifications for shelter, 822 
for tropics, 819 
wall, 818 

Tertiary plane-table triangulation, 
151 

Theodolite, angular tachymetry 
with, 272 
direction, 555 
micrometer, 554 
small exploratory, 163 
vernier, 554 
Theodolites, 76 
direction, 585 

for primary triangulation, 554 
repeating, 584, 586 [555 

of U. S. Coast and Geodetic Survey, 
of U. S. Geological Survey, 556 
Thermometers, 503 
Thermometric leveling, 402 
Thiery, E. 493 




INDEX. 


907 


Three-arm protractor, 211 
Three-point problem, 185, 600 

problem, Bessel’s solution of, 188 
problem, coast survey, solution of, 
1 go 

problem, graphic solution of, 168 
proplem, tracing-paper solution 
of, 187 

Thumb-tacks, 177 
Thwaite, B. Howarth, 884 
Tidal-fiat, 127 

Time, approximate determination 
of, from stars, 705 
approximate from sun, 703 
and arc, interconversion of, 698 
comparison of, 774 
differences, determining, 749 
determination of, 700 
estimates, 87 
Greenwich, 747 
interconversion of, 695 
intervals, relations of, 695 
mean, sidereal, solar, civil, stand¬ 
ard, etc., 683 

measurement of distances by, 226 
by meridian transits, 703 
observations, record of, 754 
reduction of, 752 
observing for, 751 
relations of sidereal to right ascen¬ 
sion and hour angle, 695 
relations of sidereal, solar, and 
civil, 695 

by single altitude, 702 
Tin cones, phase in, 560 
Tittmann, O. H., 493 
Toilet articles, 852 
Toning photographs, 880 
Tools, camp, 833 

Topographer, attributes of a skill¬ 
ful, 811 

Topographer, instructions to, 8 
Topographic agencies, no. 
control, 13 
drawing, 449 

field-work, instructions as to, 12 
forms, 72, 73, 120, 121 
forms, classification of, 122 
forms, glossary of, 133 
forms, origin and development 
of, 109 

map, 5, 39, 62, 109, 147 
map of Baltimore, 66, 107 
map, completed, 31 
map, construction of, from photo¬ 
graphs, 303 

map, correctness of, 15 


Topographic map about Cripple 
Creek, 85 

map on land survey control, 41 
map, military, 100, 101 
map about Pike’s Peak, 83 
map, public uses of, 6 
map of Yosemite Park, 115 
maps, accuracy of, 9, 15 
maps, accuracy desirable in, 8 
maps, features shown on, 5 
maps, scales for, 447 
maps, scale of detailed, 67 
maps, value of, to government, 6, 
7 , 8 

maps, value of, to State, 6, 7, S 
and plane surveying, 146 
sketching, 13, 15 
sketching, art of, 40 
survey of canal line, 55 
survey for canal location, 52 
survey of city, 64 
survey,»cost of, 40 
survey, cost of large scale, 67 
survey, elements of a, 14 
survey, object of, 5 
surveying, economy in, 22 
surveying, Geological Survey, 
method of, 20 

surveying, instruments used in, 
21 

surveying, irregular method of, 
19 

surveying, methods of, 18 
surveying, regular method of, 19 
surveying, speed in, 22 
surveys, 1, 3, 4 
surveys, prosecution of, 5 
surveys for railway location, de¬ 
tailed, 49 

Topography, military, 92 

optical illusions in sketching, 44 
photographic, 296 
for railroads, 48 
for railway location, 47 
reference works on, 490 
relation of, to geology, 108 
with stadia, 237 
Town site survey, 62 
Tracing-paper solution of three- 
point problem, 187 
Transit, 21, 52, 57, 87, 100 
adjustment of, 204 
angular tachymetry with, 272 
astronomic, and zenith telescope, 
726 

constants, 752 
engineers, 203 



908 


INDEX. 


Transit, instrument for measuring 
primary traverse, 531 
notes, example of, 208 
notes, platting with latitudes and 
departures, 212 

notes, platting with protractor, 
210 

observations, factors for reduction 
of, 758 

of star or planet, 6S2 
traversing with, 207 
Transitman on primary traverse, 
53 1 

Transits, time by meridian, 703 
Transportation, 113, 114 

on animals, cooking outfit for, 
832 

of camp by wagons, 836 
cost of, 814 

by men, cooking outfit for, 833 
repairs, 848 

of survey party in field,,813 
Traverse, 48, 64, 69 
adjustment of, 202 
adjustment < 5 f primary, to astro¬ 
nomic positions, 542 
allowance for convergence of me¬ 
ridians in primary, 539 
azimuth on primary, 535 
chain, 22 

computation of latitudes and lon¬ 
gitudes on primary, 540 
computation of primary, 538 
control, 13 

correction for check azimuths in 
primary, 539 

corrections to primary, 538 
cost, speed and accuracy of pri¬ 
mary, 536 

error of'primarv, 497 
errors in primary, 528 
instructions for primary, 533 
instruments used in, 529 
intersection from, 202 
intersections from, 24 
locations from plane-table, 26 
method of running primary, 531 
odometer, 21, 22, 24 
paced. 21, 22, 28 
permanent marks on, 532 
personnel of party on primary, 
53 i. 535 

plane-table, Gannett, 160 
with plane-table and magnetic 
needle, 197 
primary, 34, 37 
for primary control, 527 


Traverse, primary, counting tape- 
lengths on, 535 
permanent marks on, 535 
record of primary, 536 
record and reduction of primary, 
532 

roads resulting from plane-table, 
26 

secondary. 35 
sheet, 24 
stadia, 22, 24 
with stadia, 237 
surveys, 195 
table, 36, 70 

vertical angulation from, 367 
Traversing, 24 

by plane-table and deflections, 200 
with range-finder, 284 
with solar alidade, 792 
with transit, 207 

Triangles, right-angled, formulas 
for solution of, 594 
solution of plane, 596 
solution of, with slide rule, T71 
Triangulation, 58, 63, 64, 69, 87, 95 
adjustment, general conditions ap¬ 
plied to, 616 

adjustment, local conditions ap¬ 
plied to, 616 

adjustment of traverse to, 202 
astronomic checks on, 496 
control-sheet, 24 
diagram of plane-table, 25 
error of primary, 496 
errors in vertical, 370 
instruments for primary, 553 
net, 616 

observations, record of, 588 
plane-table, 12, 23, 24 
primary, 37, 545 

primary, accuracy and probable 
error of, 553-592 

primary, cost, speed and accuracy 
of, 592 

primary, figures, 548 
primary, instructions for field¬ 
work of, 590 

primary, theodolite for, 554 
reconnaissance and execution of 
plane-table, 149 

reconnaissance for primary, 546 
reduction of primary traverse to 
primary, 527 

secondary plane-table, 150 
skeleton scheme of plane-table, 

151 

signals, 559 




INDEX. 


9O9 


Triangulation, signals, transfer of 
ends of base to, 524 
stations, intervisability of, 549 
stations, platting on projection, 
436 

from tertiary sketch points, 151 
Trigonometric control, 18 
functions, 594 

functions, logarithms of, 217 
functions, natural, 275 
functions, fundamental formulas 

of, 594 
leveling, 359 
accuracy of, 361 
computation of, 368 
levels, 70 
methods, 76 
survey, 20 

Tropics, care of health in, 853 
fruit in, 855 
tents for, 819 
Tripods, plane-table, 153 
Tripod signals, 561 
Turning points, 315, 321 
Turning-point errors in leveling, 
342 

Twist in scaffolds, errors due to, 
579 . 

Two-point problem, 193 

Umlauff, Friedrich, 406 
Underwear for camping, 851 
United States Isogonic Chart of, 
222 

Valley, 115 

Van der Grinten’s projection, 413 
Van Orden, C. H., 325 
Van Ornum, J. L., 9, 241, 266, 493, 
494 

Varnish, for negatives, 881 
Variation, secular and diurnal mag¬ 
netic, 222 

Vernal equinox, 681 
Verner, Capt. Willoughby, 94, 284 
Vernier protractor, 211 
theodolites, 554 
Vertical angle, 29 
angle lines, 20, 22 
angle sight-alidade, 161, 202 
angles. 49 
angulation, 39, 361 
angulation, computation, 363 
angulation, differences of altitude 
from, 364 

angulation in sketching, 363 
angulation from traverse, 367 
angulation, control, 10 


Vertical triangulation, errors in, 
370 

Veterinary surgery, 849 
Volcanic mountain, 123 
Volcanic neck, 129 
Vulcanic ejecta, 121 
intrusions, 121 
processes, hi 
Vulcanism, 112 
Wagner-Fennell tachymeter, 280 
Wagon for moving camp, 837 
Wainright, D. B.,491 
Wall-tent with fly, 818 
specifications for, 820 
Walling, Henry F., 492, 494 
Water, altitude by boiling-point of, 
402 

color for representation of, 464 
drinking, 855 
Water-gaps, 117, 119 
Washington, survey of, 63 
Weathering, 112 
Weighted observations, 633 
Weighting, 633 
Weights, 603 

Weldon range-finder, 286, 288 
Wellington, A. M., 40, 44, 49, 494 
Wharton, Capt. W. J. L., 810 
Wheel measurement, 229 
revolutions, counting, 232 
Wheeler, Capt. Geo. M., 9, 494 
map, 80 
survey, 73, 75 

Williamson, Lieut.-Col. R. S., 49.1 
Williamson’s barometric tables, 37 ~j 
Wilson, H. M., 494 

vertical angle sight-alidade, 161 
Wind-gaps, 119 

Winds, effect of, on leveling, 341 
Winslow, Arthur, 249, 494 
Winston, Isaac, 494 
Wires, stadia, 239 
Wisner, George Y., 810 
Witness marks, 575 
Wood for fire, 830 
Wooden tent-floor, 826 
Woodland sketching, 35 
surveying, 33 

Woods, color for representation of, 
464 

Woodward, R. S., 258,418, 494, 501 
578, 702. 810 

iced-bar base-apparatus, 511 
Works of reference on geodesy 
and astronomy, 809 
of reference on topography, 490 
Wounds, 861 





OIO 


INDEX . 


Wright, T. A., 604, 810 
on probabilities, 614, 617 
Wye level, engineers’, 310 

Yard, English equivalents of, 673 
metric equivalents of, 675 
Young & Sons, solar, 781 

Zenith, 681 


Zenith distances, differences of, for 
latitude, 723 

distances, latitude by differences 
of, 728 

distances, reciprocal at two sta¬ 
tions, 360 

telescope, 723 

telescope and astronomic transit, 
726 



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